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COPntlGHT DEPOSIT. 



WORKS OF PROF. WM. H. BURR 

PUBLISHED BY 

JOHN WILEY & SONS, Inc. 



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XV +476 pages, 6 by 9, profusely illustrated, including 
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Elasticity and Resistance of Materials of Engi- 
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For the use of Engineers and Students. Containing the 
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Cements, Mortars and Concretes— Their Physical 
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THE 



ELASTICITY AND RESISTANCE 



OF THE 



MATERIALS OF ENGINEERING. 



BY 



WM. H. BURR, C.E., 



PROFESSOR OF CIVIL ENGINEEKING IN COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK? 

CONSULTING engineer; MEMBER OF THE AMERICAN SOCIETY OF CIVIL ENGINEERS; 

MEMBER OF THE INSTITUTION OF CIVIL ENGINEERS OF 

GREAT BRITAIN. 



SEVENTH EDITION, THOROUGHLY REVISED 
TOTAL ISSUE, SEVEN THOUSAND 



NEW YORK 

JOHN WILEY & SONS, Inc. 

London: CHAPMAN & HALL, Limited 

191S 






Copyright, 1883, 1903. IQIS, 

BY 

WM. H. BURR. 

Copyright renewed, 191 1, 
BY 

WM. H. BURR. 



OCT I9I9I5 



ro^ 



CI,A416001 
'VVO- \ • 



PREFACE TO SEVENTH EDITION. 



The rapid development which has characterized all 
branches of engineering construction during the past 
decade carries with it corresponding advances in experi- 
mental and analytic work in that field of engineering 
science known as the Elasticity and Resistance of Mate- 
rials. In the present edition of this book, prepared to 
meet the advancing requirements of the profession, it 
will be observed that much of the older matter has been 
canceled and displaced by many new topics now become 
of practical importance, so that new material constitutes 
probably not less than three-quarters of the volume. These 
new parts will readily be discovered by a glance at the 
contents. It may be well, however, to state that the 
treatment of reinforced concrete, the general analysis of 
which as a development of the common theory of flexure 
was first given in a prior edition of this book, has been 
extended to cover substantially all the principal features 
of that special field. The analysis given is general, 
but simple and free from the superfluous and labor- 
increasing accretions which, for some not obvious reasons, 
have found place in some of the commonly used formulae. 

Results of the most recent experimental investigations 
have been used for the requisite empirical data, so as to 
make the book a real work on the Elasticity and Resist- 
ance of the Materials of Engineering rather than a mere 
matter of applied mechanics. 

W. H. B. 

Columbia University, 
Oct. 1,1915. 



CONTENTS. 



PART I. 
ANALYTICAL. 

CHAPTER I. 

ELEMENTARY THEORY OF ELASTICITY IN AMORPHOUS SOLID 

BODIES. 

ART. PAGE 

1 . General Statements i 

2. Coefficient or Modulus of Elasticity 4 

3. Direct Stresses of Tension and Compression. . . .- 7 

4. Lateral Strains 9 

5. Relation between the Coefficieilts of Elasticity for Shearing and 

Direct Stress in a Homogeneous Body 11 

6. Shearing Stresses and Strains 13 

7. Relation between Moduli of Elasticity and Rate of Change of 

Volume 18 

8. All Stresses Parallel to One Plane — Resultant Stress on any Plane 

Normal to the Plane of Action of the Stresses 21 

Sum of Normal Components 24 

9. The Ellipse of Stress — Greatest Intensity of Shearing Stress — 

Equivalence of Pure Shear to Two Principal Stresses of Opposite 
Ednds but Equal Intensities — Greatest Obliquity of Resultant 

Stress on any Plane 26 

Greatest Intensity of Shearing Stress 29 

Equivalence of Pure Shear to Tico Principal Stresses of Opposite 

Kinds but Equal Intensities 30 

Greatest Obliquity of Resultant Stress on any Plane 31 

10. Ellipse of Stress and Resulting Formulas for the Special Case of 

Zero Intensity of One of the Known Direct Stresses. ..,..■ 33 

1 1 . General Condition of Stress — Ellipsoid of Stress 36 

Principal Stresses and Ellipsoid of Stress 40 

vii 



VIU CONTENTS. 

ART. PAGE 

12. Ellipse and Ellipsoid of Strain ' 43 

13. Orthogonal Stresses 43 

CHAPTER II. 
FLEXURE. 

14. The Common Theory of Flexure 49 

15. The Distribution of Shearing Stress in the Normal Section of a Bent 

Beam 57 

Distribution of Shear in Circular and Other Sections 62 

16. External Bending Moments and Shears in General • 64 

17. Intermediate and End Shears 68 

18. Maximum Reactions for Bridge Floor Beams 74 

19. Greatest Bending Moment Produced by Two Equal Weights 76 

20. Position of Uniform Load for Greatest Shear and Greatest Bending 

Moment at any Section of a Non-continuous Beam — ^^Bending 
Moments of Concentrated Loads 79 

21. Greatest Bending Moment in a Non-continuous Beam Produced by 

Concentrated Loads 83 

22. Moments and Shears in Special Cases 94 

Case 1 95 

Case II 96 

Case III 98 

23. Recapitulation of the General Formulae of the Common Theory of 

Flexure 99 

24. The Theorem of Three Moments 102 

25. Short Demonstration of the Common Form of the Theorem of 

Three Moments 114 

26. Reaction under Continuous Beam of any Number of Spans 118 

27. Deflection by the Common Theory of Flexure 121 

Deflection Due to Shearing 125 

28. The Neutral Curve for Special Cases ; 126 

Case I 126 

Case II 129 

Case III 131 

Addendum to Art. 28 143 

29. Direct Demonstration for Beam Fixed at One End and Simply Sup- 

ported at the Other under Uniform and Single Loads 144 

Special Case, a = \ 149 

30. Direct Demonstration for Beams Fixed at Both Ends under Uniform 

and Single Loads 150 

31. Deflection Due to Shearing in Special Cases 153 

32. The Common Theory of Flexure for a Beam Composed of Two 

Materials , 156 



CONTENTS. ix 

ART. PAGE 

33. Graphical Determination of the Resistance of a Beam 160 

34. Greatest Stresses at any Point in a Beam 162 

35. The Flexure of Long Columns 169 

36. Special Cases of Flexure of Long Columns 175 

Flexure by Oblique Forces • 175 

Column Free at Upper End and Fixed Vertically at Lower End 
with either Inclined or Vertical Loading at Upper End 177 

CHAPTER III. 
TORSION. 

37. Torsion in Equilibrium 182 

Twisting Moment in Terms of Horse-power H 188 

Hollow Circular Cylinders 189 

38. Practical Applications of Formulae for Torsion 190 

Steel 190 

Wrought Iron 192 

Cast Iron 192 

Alloys of Copper, Tin, Zinc and Aluminum 193 

Other Sections than Circular 1-96 

CHAPTER IV. 
HOLLOW CYLINDERS AND SPHERES. 

39. Thin Hollow Cylinders and Spher,es in Tension 197 

40. Thick Hollow Cylinders 203 

Case of Exterior Pressure Greater than Interior Pressure 211 

41. Radial Strain or Displacement in Thick Hollow Cylinders — 

Stresses Due to Shrinkage of One Hollow Cylinder on Another. . . 212 

Radial Strain or Displacement. 212 

Stresses Due to Shrinkage 213 

Inner Cylinder in Compression 217 

Outer Cylinder in Tension ■ 218 

Combined Cylinder under High Internal Pressure 219 

42. Thick Hollow Spheres 224 

Radial Displacement at any Point in the Spherical Shell 230 

CHAPTER V. 

RESILIENCE. 

43. General Considerations 231 

44. The Elastic Resilience of Tension and Compression and of Flexure. 232 

The Resilience of Bending or Flexure 233 



X CONTENTS. 

ART. PAGE 

The Resilience Due to the Vertical or Transverse Shearing Stresses 

in a Bent Beam 236 

The Total Resilience Due to Both Direct ajid Shearing Stresses . . . 239 

45. Resilience of Torsion 240 

46. Suddenly Applied Loads 242 



CHAPTER VI. 
COMBINED STRESS CONDITIONS. 

47. Combined Bending and Torsion 246 

First Method 248 

Second Method 250 

48. Combined Bending and Direct Stress 254 

49. The Eye-bar Subjected to Bending by Its Own Weight or Other 

Vertical Loading 255 

Approximate Method 256 

50. The Approximate Method Ordinarily Employed 258 

5 1 . Exact Method of Treating Combined Bending and Direct Stress. . . . 263 

52. Combined Bending and Direct Stress in Compression Members 268 

Exact Method for Combined Compression and Bending 271 



PART II. 
TECHNICAL, 

CHAPTER VII. 
TENSION. 

53. General Observations. — Limit of Elasticity. — Yield Point 281 

Yield Point 284 

54. Ultimate Resistance 285 

55. Ductility — Permanent Set 286 

56. Cast Iron 286 

Modulus of Elasticity and Elastic Limit 286 

Resilience, or Work Performed in Straining Cast Iron 290 

Ultimate Resistance 292 

Effects of Remelting, Continued Fusion, Repetition of Stress, and 

High Temperature, 294 



CONTENTS. XI 

ART. PAGE 

57. Wrought Iron — Modulus of Elasticity — Limit of Elasticity . and 

Yield Point — Resilience — Ultimate Resistance and Ductility .... 295 

Modulus of Elasticity 296 

Limit of Elasticity and Yield Point Resilience 297 

Ductility and Resilience 299 

Ultimate Resistance 301 

Ductility 302 

Fracture of Wrought Iron 302 

58. Steel 303 

Modulus of Elasticity 303 

Variation of Ultimate Resistance with A rea of Cross-section 308 

Influence of Shortness of Specimen 309 

Elastic Limit, Resilience, and Ultimate Resistance 310 

Shape Steel and Plates 315 

Carbon Steel for Towers 318 

Carbon Steel for Suspended Structures 319 

Nickel Steel for Stiffening Trusses 319 

Steel Wire 320 

Steel Castings 321 

Rail Steel 323 

Rivet Steel 324 

Nickel Steel 325 

Vanadium Steel 328 

Effect of Low and High Temperatures 333 

Hardening and Tempering. . .' 336 

Annealing 338 

Effect of Manipulations Common to Constructive Processes; 

also Punched, Drilled and Reamed Holes 339 

Change of Ultimate Resistance, Elastic Limit and Modulus of 

Elasticity by Retesting 342 

Fracture of Steel 343 

The Effects of Chemical Elements on the Physical Qualities of 

Steel 343 

59. Copper, Tin, Aluminum, and Zinc, and Their Alloys — Alloys of 

Aluminum — Phosphor-Bronze — Magnesium 346 

Ultimate Resistance and Elastic Limit 348 

Alloys of Aluminum 352 

Alloys of Aluminum and Copper 357 

Bronzes and Brass Used by the Board of Water Supply of New 

York City 359 

Phosphor-Bronze 361 

Bauschinger's Tests of Copper and Brass as to Effect of Repeated 
Application of Stress 361 



XU CONTENTS. 

ART. PAGE 

60. Cement, Cement Mortars, etc. — Brick 362 

Modulus of Elasticity 363 

Ultimate Resistance 365 

Weight of Concrete 372 

Adhesion between Bricks and Cement Mortar 373 

The Effect of Freezing Cements and Cement Mortars 375 

The Linear Thermal Expansion and Contraction of Concrete and 
Stone 377 

61 . Timber in Tension 379 



CHAPTER VIII. 
COMPRESSION. *» 

62. Preliminary 385 

63. Wrought Iron 387 

Modulus of Elasticity 387 

Limit of Elasticity and Ultimate Resistance .- 388 

64. Cast Iron 388 

65. Steel ; 389 

66. Copper, Tin, Zinc, Lead, and Alloys 391 

67. Cement — Cement Mortar — Concrete . 395 

68. Bricks and Brick Piers 409 

Brick Piers 413 

69. Natural Building Stones 420 

70. Timber 426 



CHAPTER IX. 
RIVETED JOINTS AND PIN CONNECTION. 

7 1 . Riveted Joints 435 

Kinds of Joints 435 

72. Distribution of Stress in Riveted Joints 437 

Bending of the Plates 437 

Net Section of Plates 439 

Bending of the Rivets 440 

The Bearing Capacity of Rivets 441 

Bending of Plate Metal in Front of Rivets 442 

Shearing of Rivets. . .• 443 



CONTENTS. xiii 

ART. PAGE 

73. Diameter and Pitch of Rivets and Overlap of Plate. — Distance 

between Rows of Riveting 445 

Diameter of Rivets 445 

Pitch of Rivets 446 

Overlap of Plate 447 

Distance between Rows of Riveting 448 

74. Lap-joints, and Butt-joints with Single Butt-strap for Steel 

Plates ' 448 

75. Steel Butt-joints with Double Cover-plates 452 

76. Tests of Full-size Riveted Joints 454 

Efficiencies 461 

77. Tests of Joints for the American Railway Engineering and Main- 
. . tenance of Way Association and for the Board of Consulting 

Engineers of the Queb^ Bridge 462 

Friction of Riveted Joints 465 

78. Riveted Truss Joints , 467 

Diagonal Joints 469 

Riveted Joints in Angles 469 

Hand and Machine Riveting 470 

79. Welded Joints 470 

80. Pin Connections .,,,,,,., 470 



CHAPTER X. 
LONGCOLUMNS. 

81. Preliminary Matter 474 

Principal Momejits of Inertia 477 

82. Gordon's Formula for Long Columns 481 

83. Tests of Wrought-iron Phoenix Columns, Steel Angles and Other 

Steel Columns 490 

Steel Columns 496 

Typical Formulce Now in Use 503 

Details of Columns 505 

84. Complete Design of Pin-end Steel Columns 506 

85. Cast-iron Columns 520 

86. Timber Columns 528 

Formula of C. Shaler Smith 531 

Tests of White Pine and Yellow Pine Full-size Sticks with Flat 
Ends 533 



XIV CONTENTS. 

CHAPTER XI. 
SHEARING AND TORSION. 

ART. PAGE 

87. Modulus of Elasticity 540 

88. Ultimate Resistance '. 543 

Wrought Iron 543 

Cast Iron 544 

Steel "545 

Copper, Tin, Zinc, and Their Alloys 546 

Timber 547 

Natural Stones 549 

Bricks 550 

CHAPTER XII. 
BENDING OR FLEXURE. 

89. Modulus of Elasticity 552 

90. Formulae for Rupture 552 

91. Beams with Rectangular and Circular Sections 554 

High Extreme Fibre Stress in Short Solid Beams 556 

Steel .558 

Cast Iron 560 

Alloys of Aluminum 560 

Copper, Tin, Zinc, and their Alloys 561 

Timber Beams 563 

Failure of Timber Beams by Shearing along the Neutral Surface. 571 

Influence of Time on the Strains of Timber Beams 574 

Concrete Beams 575 

Natural-stone Beams 586 



CHAPTER XIII. 
CONCRETE-STEEL MEMBERS. 

92. Composite Beams or Other Members of Concrete and Steel 588 

■ 93. Physical Features of the Concrete-steel Combination in Beams 589 

94. Rate at which Steel Reinforcement Acquires Stress 592 

95. Ultimate and Working Values of Empirical Quantities for *Concrete- 

steel Beams 598 

96. General Formulae and Notation for the Theory of Concrete-steel 

Beams according to the Common Theory of Flexure 600 



CONTENTS. XV 

ART. PAGE 

97. T-beams of Reinforced Concrete 604 

Position of Neutral Axis 605 

Balanced or Economic Steel Reinforcement 608 

Formula to Locate Neutral Axis in T-beams 610 

98. Bending Moments in Concrete-steel T-beams by Common Theory 

of Flexure 614 

Neglect of Concrete in Tension 615 

Special Case of Neutral Axis in under Surface of Flange 616 

99. Concrete Steel Beams of Rectangular Section 616 

FormulcB to Locate Neutral Axis in Beams of Rectangular Section 616 

Bendijig Moments for Rectangular Sections 618 

Neglect of Concrete in Tension 619 

100. Shearing Stresses and Web Reinforcements in Reinforced Concrete 

Beams. . , 620 

lOi. Working Stresses and Other Conditions in Reinforced Concrete 

Designs-Design of T-beams 629 

Working Stresses '. 631 

Working Compression in Extreme Fibre of Beam 631 

Shear and Diagonal Tension 632 

Bond or A dhesive Shear 633 

Steel Reinforcement 633 

Modulus of Elasticity 633 

Design of T-beam for Heavy Uniform Load 634 

Design of Continuous Floor- Slab for 6-foot Spans 639 

102. Reinforced Concrete Columns 641 

Lateral Reinforcement and Shrinkage 642 

Longitudinal Reinforcement 644 

Types of Columns 646 

Working Stresses 650 

103. Division of Loading between the Concrete and Steel under the 

Common Theory of Flexure 655 



CHAPTER XIV. 
ROLLED AND CAST-FLANGED BEAMS. 

104. Flanged Beams in General 659 

105. Flanged Beams with Unequal Flanges 661 

106. Flanged Beams with Equal Flanges : 665 

107. Rolled Steel Flanged Beams 669 

108. The Deflection of Rolled Steel Beams 677 

109. Wrought-iron Rolled Beams 679 



xvm CONTENTS. 

APPENDIX I. 

ELEMENTS OF THEORY OF ELASTICITY IN AMORPHOUS 
SOLID BODIES. 



CHAPTER I. 
GENERAL EQUATIONS. 

ART. PAGE 

1 . Expressions for Tangential and Direct Stresses in Terms of the Rates 

of Strains at Any Point of a Homogeneous Body 820 

2. General Equations of Internal Motion and Equilibrium 826 

3. Equations of Motion and Equilibrium in Semi-polar Co-ordinates. . . 832 

4. Equations of Motion and Equilibrium in Polar Co-ordinates 839 



CHAPTER II. 
THICK, HOLLOW CYLINDERS AND SPHERES, AND TORSION. 

5. Thick, Hollow Cylinders 847 

6. Torsion in Equilibrium 853 

Equations of Condition in Rectangular Co-ordinates 860 

Solutions of Eqs. (13) and (21) 862 

Elliptical Section about its Centre 863 

Equilateral Triangle about its Centre of Gravity 866 

Rectangular Section about an Axis Passing through its Centre 

of Gravity 869, 883 

Square Section 878, 882 

Greatest Intensity of Shear 880 

Circular Section about its Centre. . 884 

General Observations 885 

7. Torsional Oscillations of Circular Cylinders 886 

8. Thick, Hollow Spheres 892 

CHAPTER III. 

THEORY OF FLEXURE. 

9. General Formulae 897 



CONTENTS. xix 

APPENDIX 11. 

Page 
CLAVARINO'S FORMULA 913 



APPENDIX III. 

RESISTING CAPACITY OF NATURAL AND 

ARTIFICIAL ICE 916 

Index 921 



ELASTICITY AND RESISTANCE 
OF MATERIALS. 



PART I.— ANALYTICAL 



CHAPTER I. 

ELEMENTARY THEOPY OF ELASTICITY IN AMORPHOUS 
SOLID BODIES. 

Art. I. — General Statements. 

The molecules of all solid bodies known in nature are 
more or less free to move toward, or from, or among each 
other. Resistances are offered to such motions, which 
vary according to the circumstances under which they 
take place and the nature of the body. This property 
of resistance is termed the elasticity of the body. 

The summation of the displacements of the molecules 
of a body, for a given point, is called the distortion or 
strain at the point considered. The force by which the 
molecules of a body resist a strain, at any point, is called 
the stress at that point. This distinction between stress 
and strain is fundamental and important. 

Stresses are developed, and strains caused, by the 
application of force to the exterior surface of the material. 



2 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

These stresses and strains vary in character according to 
the method of appHcation of the external forces. Each 
stress, however, is accompanied by its own characteristic 
strain and no other. Thus there are shearing stresses and 
shearing strains, tensile stresses and tensile strains, com- 
pressive stresses and compressive strains. Usually a 
number of different stresses with their corresponding 
strains are coexistent at any point in a body subjected to 
the action of external forces. 

It is a matter of experience that strains always vary 
continuously and in the same direction with the corre- 
sponding stresses. Consequently the stresses are con- 
tinuously increasing functions of the strains, and any 
stress may be represented by a series composed of the 
ascending powers (commencing with the first) of the strains 
multiplied by proper coefficients. When, as is usually 
the case, the displacements are very small, the terms of 
the series whose indices are greater than unity are ex- 
ceedingly small compared w4th the first term, whose index 
is unity. Those terms may consequently be omitted 
without essentially changing the value of the expression. 
Hence follows what is ordinarily termed Hooke's law: 

The ratio between stresses and corresponding strains, jot 
a given material, is constant. 

This law is susceptible of very simple algebraic repre- 
sentation. If a piece of material, whose normal cross- 
section is A, is subjected to either tensile or compressive 
stress, its length L will be changed by the amount JL. 
If P be the external force or loading which produces that 
deformation or change of length, the amount of force or 
stress, supposed to be uniformly distributed, acting on i 
square inch of normal cross-section of the piece, will be 
found by dividing the total force P by the area of cross- 
section A. This amount of uniformly distributed stress 



Art. I.] GENERAL STATEMENTS. 3 

is called the ' ' intensity of stress, ' ' and it is a most impor- 
tant quantity. In dealing with the effects of forces or 
stresses in all engineering work, the amount of such force 
or stress on a square unit of area, usually a square inch in 
American practice, and called the intensity, is often the 
main object sought, for it determines the question whether 
material is carrying too much or too little load, as well as 
many other related questions. 

Again, the important consideration as to strain is the 
fractional change in length of the entire piece, and not the 
total change in length expressed in the unit adopted, ordi- 
narily an inch. This fractional change of length is the same 
as the amount of actual change of each linear unit of the 
piece, as found by dividing JL by L. Inasmuch as that 
fraction expresses the amount of change in length for each 
imit, it is frequently called the rate of change of length or 
rate of deformation. Hooke's law^ is to the effect that 
the intensity of stress is proportional to the rate of strain, 
and its analytic expression may readily be written. 

Let p represent the intensity of any stress and / the 
strain per unit of length, or, in other words, the rate of 
strain. If £ is a constant coefficient, Hooke 's law will be 
given by the following equation: 

P AL 

If the intensity of stress varies from point to point of a 
body, Hooke's law may be expressed by the following 
differential equation: 

■§=^^ • ■ « 

If p and I are rectangular coordinates, eqs. (i) and (2) 
are evidently equations of a straight line passing through 



4 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

the origin of coordinates. It will hereafter be seen that 
the line under consideration is essentially straight for 
comparatively small strains in any case, and for some 
materials it has no straight portions. 

Art. 2. — Coefficient or Modulus of Elasticity. 

In general the coefficient E in eq. (i) of the preced- 
ing article is called the coefficient of elasticity, or, more 
usually, modulus of elasticity. The coefficient of elasticity 
varies both with the kind of material and kind of stress. 
It simply expresses the ratio between the rates of stress and 
strain. 

The characteristic strain of a tensile stress is evidently 
an increase of the linear dimensions of the body in the 
direction of action of the external forces. 

Let this increase per unit of length be represented by 
/, while p and E represent, respectively, the correspond- 
ing intensity and coefficient. Eq. (i) of the preceding 
article then becomes 

p=El, or £=1 (i) 

E is then the coefficient of elasticity for tension. 

The characteristic strain for a compressive stress is 
evidently a decrease in the linear dimensions of the body 
in the direction of action of the external forces. Let l^ 
represent this decrease per unit of length, p^ the intensity 
of compressive stress, and E^ the corresponding coefficient. 
Hence 

/>.=£,?, or £,= ! (2) 

E^ consequently is the coefficient of elasticity for 
compression. 



Art. 2.] 



COEFFICIENTS OF ELASTICITY, 



The characteristic strain for a shearing stress may be 
determined by considering the effect which it produces 
on the layers of the body parahel to its plane of action. 

In Fig. I let A BCD represent one face of a cube, another 
of whose faces is fixed along AD. If a shear acts in the 
face BC, whose plane is normal to the plane 
of the paper, all layers of the cube parallel 
to the plane of the shearing stress, i.e., BC, 
will slide over each other, so that the faces 
AB and DC will take the positions AE and 
DF. The amount of distortion or strain 
per unit of length will be represented by 
the angle EAB = (p. If the strain is small, 
there may be written ^, sin ^, or tan (f> 
indifferently. 

Representing, therefore, the intensity of shear, coeffi- 
cient, and strain by 5, G, and (f), respectively, eq. (i) of 
Art. I becomes 




S=G(l), or G 



S 



(3) 



It will be seen hereafter that there are certain limits 
of stress within which eqs. (i), (2), and (3) are essentially 
true, but beyond which they do not hold; this limit is 
called the limit of elasticity, and is not in general a well- 
defined point. 

The line Okghn exhibited in Fig. 2 represents the actual 
strains in a piece of structural steel i inch in length with 
I square inch of cross-section. is the origin- of coordi- 
nates, and the loads per square inch, i.e., intensities of 
stresses, are shown by the vertical ordinates drawn parallel 
to OC from OD to the strain curve, while the strains per 
unit of length, that is, per inch, are laid off as horizontal 
ordinates of the curve parallel to OD. If Op' is the in- 



6 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

tensity of stress, p' corresponding to the point k of the strain 
curve, while 01' is the resulting strain per unit of length, 
then p' =EU. Again, if g is at the upper limit of the straight 
portion of the curve for which the intensity of stress and 
rate of strain are p and / respectively, the relation between 
those two quantities is shown by eq. (i). Since E, also as 




Fig. 2. 

shown by eq. (i), is equal to the quotient of p divided 
by /, Fig. 2 shows that it is equal to the tangent of the 
angle between OD and the straight portion Og of the strain 
curve, it being supposed that the rates of strain are laid 
down at their actual or natural sizes. If the strain line is 
curved, the first term of eq. (2) of Art. i, the differential 
ratio, w411 represent the tangent of the angle between the 
curve and the horizontal axis OD in Fig. 2. The point g, 
being at the upper limit of constant proportionality be- 
tween intensity of stress and rate of strain, is called the 
elastic Hmit, above which it is seen that the strains in- 
crease far more rapidly than the stresses until the point n 
is reached, where actual rupture takes place. The nearly 
horizontal portion of the curve between g and h and a little 



Art. 3.] STRESSES OF ThNSION AND COMPRESSION. 7 

above g indicates the " yield point," an. intensity of 
stress where the material is said first to * ' break down ' ' or 
stretch rapidly under tensile stress without much increase 
of the latter. 



Art. 3. — Direct Stresses of Tension and Compression. 

The direct stresses of tension and compression always 
produce shearing stresses and strains on all planes in the 
interior of a body except those perpendicular and parallel 
to those direct stresses. If, in Fig. i, a straight piece of 
material CD is subjected to the tensile stress induced by 
the forces P equal and opposite to each other, there will be 
pure tension only on all planes or sections of the piece at 
right angles to the direction of the forces P, such as HK. 
On all planes passing through the longitudinal axis of the 
piece there will be no stress whatever, if, as is supposed, 
the forces P are uniformly distributed over the sections 
of application DF and BC. 




On every oblique plane or section in all parts of the 
piece as H'K\ supposed to be perpendicular to the plane 
of the diagram, there will be shear as well as direct 
stress of tension normal to it, the intensities of both the 
shear and the normal stress being dependent upon the 
angle a between HK and H^K\ The force P may be 
resolved by the triangle of forces into two components, 
one at right angles to H'K' , represented by A^, and the 
other along or tangential to H^K\ represented by 5. If 



8 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

A represents the area of the normal section HK, the area 

of the obhque section H^K^ will be A sec a. The value 

of the noiTnal stress A^ will be N =P cos a, but S =P sin a. 

The intensity of the normal tensile stress on H^K' will be, 

therefore, 

N P cos a: , 

n=-, = -. =pcos^a. . . (i) 

A sec a A sec a ^ ^ ^ 

The intensity of shear on the same plane H^K^ will be 

S P sin a . 

s=-, =-. =p sm a cos a. . . (2) 

A sec a A sec a ^ ^ 

"VVnen the angle a is zero, 5 in eq. (2) becomes zero, 
while n in eq. (1) becomes equal to p, i.e., the intensity of 
direct tensile stress on the normal section. On the other 
hand, when the angle a has the value of 90°, both n and 5 
become zero, i.e., there is no stress whatever on a longi- 
tudinal, axial plane. 

Inasmuch as the angle a may have any value w^hat- 
ever from zero to 90° on either side of HK, it is clear that 
both shearing and normal tensile stresses will be found 
concurrently on every oblique plane in the piece. As has 
been observed in the preceding article, these shearing 
stresses induce the lateral strains under which the normal 
cross-sections of a piece subjected to pure tension decrease 
in area \vhile they increase under the action of pure com- 
pression. 

Eqs. (i) and (2) have been written on the assump- 
tion that the external forces P produce tension in the 
material, but precisely the same equations apply to the 
condition of pure comxpression, the only difference being 
that in the latter case the external forces P w^ould be di- 
rected toward each other from the ends of the piece, in- 
stead of away from each other. 



Art. 4.] LATERAL STRAINS. 



Art. 4. — Lateral Strains. 

If a body, as indicated in Fig. i, be subjected to ten- 
sion, it .has been shown in Art. 3 that all of its oblique cross- 
sections, such as FE and GH, will sustain shearing stresses 
in consequence of the component of the tension tangential 
to those oblique sections. These tangential stresses will 
cause the oblique sections, in both directions, to slide over 

AGE c 



'Ma' 



>-<^^•^ 



-P^ 



H 

Fig. 1 



each other. Consequently the normal cross-sections of the 
body will be decreased; and if the norma] cross-sections of 
the body are made less, its capacity to resist the external 
forces acting on AB and CD will be correspondingly dimin- 
ished. 

If the body is subjected to compression, oblique sec- 
tions of the body will be subjected to shears, but in direc- 
tions opposite to those existing in the previous case. The 
effect of such shears will be an increase of the lateral 
dimensions of the body and a corresponding increase in 
its capacity of resistance. 

These changes in the lateral dimensions of the body are 
termed ''lateral strains"; they always accompany direct 
strains of tension and compression. 

It is to be observed that lateral strains decrease a body 's 
resistance to tension, but increase its resistance to com- 
pression. Also, that if they are prevented, both kinds of 
resistance are increased. 

Consider a cube, each of whose edges is a, in a body 
subjected to tension. Let r represent the ratio between 



lo . ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

the lateral and direct strains,* and let it be supposed to 
be the same in all directions. If Z, as in Art. 2, represents 
the direct unit strain, the edges of the cube will become, by 
the tension, a{i+l), a(i—lr), and a{i—rl). Consequently 
the volume of the resulting parallelopiped will be 

a\i+l){i-riy^a\i-\-l{i-2r)] . .. . (i) 

if powers of / higher than the first be omitted. With r be- 
tween o and i, there will he an increase oj vo!:i;ne, but not 
otherwise. 

If the body is subjected to compression, the edges of 
the cube become a(i— /J, a(i+r^/J, and a{i+rj^)] while 
the volume of the parallelopiped takes the value 

a\i-lj(i+r,l,r=a\i+l,(2r^-i)l . . (2) 

As before, the higher powers of l^ are omitted. If the 
volume of the cube is decreased, r^ must be found between 
o and J. 

If a be unity in eq. (i), it is then clear that the expres- 
sion l(i — 2r) is the change of volume of a unit cube, i.e., 
it is the rate of change of volume w^hen the intensity of stress 
is p=El. Hence if this rate of change of volume be mul- 
tiplied by a definite volume V the result will be the total 
change of that definite volume produced by the uniform 
intensity of stress p. 

If the intensity of stress varies from point to point the 
total change of volume will become : 



/!<■- "'■"'= (t)/^""- 



(3) 



Evidently the volume V must be expressed in the same 
independent variable, or variables, as p. The integral 
must then be made to cover the desired limits. 
* Frequently called Poisson's ratio. 



Art. 5.] RELATION BETIVEEN COEFFICIENTS OF ELASTICITY- 



Art. 5. — Relation between the Coefficients of Elasticity for 
Shearing and Direct Stress in a Homogeneous Body. 

A body is said to be homogeneous when its elasticity, 
of a given kind, is the same in all directions. 

Let Fig. I represent a body subjected to tension parallel 
to CD. That oblique section on which the shear has the 
f^ £ B greatest intensity will make 

an angle of 45° with either of 
those faces whose traces are 
CD or BD ; for if a is the angle 
which any oblique section 
"D makes with BD, P the total 
tension on BD, and A' the 
area of the latter surface, the total shear on any section 
whose area is A' sec a will be P sin a. Hence the intensity 
of shear is 

P sin a P . 



E>1><i\ 


/ 


-d^iT—HZi 1- IfL- _ jk/ 


/k 








\^ 










"^^ X I'' / 


^y 






G'-JXt/ 


\ 



G 

Fig. I, 



-77 = T, vSm a cos a. 

A sec a A 



(i) 



The second member of eq. (i) evidently has its greatest 

value for a =45°. Hence if the tensile intensity on BD is 

P 
represented by -r-, =p, the greatest intensity of shear will be 

P 



S = 



Then by eq. (3) of Art. 2, 



2G' 



(2) 



(3) 



In Fig. I EK and KG are perpendicular to each other, 
while they make angles of 45° with either AB or CD. After 
stress, the cube EKGH is distorted to the oblique paral- 
lelepiped E'KG'H'. Consequently EKGH and E'KG'H' 
correspond to A BCD and AEFD, respectively, of Fig. i, 



12 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

Art. 2. The angular difference EKG — E' KG' is then equal 
to ^ ; and EKE' = GKG' ^ ^. Also, E'KF' = 4S° - - 

2 2 

Using, then, the notation of the preceding articles, 
there will result, nearly, 

tan(45°-^) = ^^ = i-/(i+^); • . (4) 

remembering that 

F'K=FK{i+l), and E'F' =FK(i -rl). 

From a trigonometrical formula there is obtained, 
very nearly, 

,\ tan4S° — tan— i — — 



tan^45°-jj-^ ^.__=,_^. . (5) 

tan 45°+ tan— i + — 
^22 

From eqs. (4) and (5), 

(/>=/(! +r) (6) 

Substituting from eq. (3), as well as from eq. (i) of 
Art. 2, 

E 

^^^+7) (^^ 

It has already been seen in the preceding article that r 
must be found between o and ^, consequently the coefficient 
of elasticity for shearing lies between the values of \ and \ of 
that of the coefficient of elasticity for tension. 

This result is approximately verified by experiment. 

Since precisely the same form of result is obtained by 
treating compressive stress, instead of tensile, there will be 
found, by equating the two values of G, 

E E, E, i-\- r, 

' or 1=^=---' (8) 



I + r I + r/ E I + r 



Art. 6.J SHEARING STRESSES AND STRAINS. 13 

It is clear, from the conditions assumed and operations 
involved, that the relations shown by eqs. (7) and (8) can 
only be approximate. 

Art. 6. — Shearing Stresses and Strains. 

In the preceding Arts, the more simple and ordinary 
relations between stress and strain are shown, but in this 
and follow^ing Arts, it is desirable to give a more extended 
treatment. 

Materials are rarely used in structures and machines 
under conditions in which the stress is wholly shear. The 
usual conditions are such as to produce shear concurrently 
with stresses of tension and compression. Even in the use 
of rivets, where shearing stress acts prominently, tension 
and compression in the form of flexure and direct com- 
pression are concurrent. Again in the case of flexure or 
the bending of beams, the shearing stress is sufficiently 
high in intensity in some cases to produce failure, but 
concurrently with relatively' high intensities of tension 
and compression. 

Figs. I and 2 show a rectangular parallelopiped of 
material of depth b at right angles to the plane ABCD 
firmly held on the face AD, while the intensities of shear 
s and s^ act on the faces AB, EC, CD, and AD. It is 
supposed that no other stresses act upon the exterior 
faces of the prism of material. Let the prism be imagined 
to be divided into indefinitely thin vertical slices at right 
angles to the face A BCD when in its original position 
shown by AB'C'D. Similarly let the prism be imagined 
to be divided into indefinitely thin horizontal slices at 
right angles to the same face. 

Before considering the distortion of the prism due to 
the action of the shearing stresses an important but simple 
principle must be established. As there are no stresses 



14 



ELASTICITY IN AMORPHOUS SOLID BODIES. 



[Ch. I. 



acting upon the prism except the opposite pairs of shearing 
stresses whose intensities are 5 and s' as shown, it is clear 
that the prism must be held in equilibrium by the two 
couples acting in opposite directions whose lever arms 
are AB' and AD. Let / represent the length AB' of the 





Fig. 



Fig. 2. 



prism, while AD=d, as shown in Fig. 2. Then since the 
prism is in equilibrium there will result the equation, 



s'hl.d=shd.l 



(i) 



This equation shows that the intensities of two shears 
acting on planes at right angles to each other and parallel 
to a third plane at right angles to the other two must be 
equal. Furthermore, it is clear from Fig. 2 that the shears 
on the faces of the prism must act in pairs toward two of 
the corners of the prism diagonally opposite each other 
and away from the other diagonally opposite pair of corners. 

The rectangular prism of Figs, i and 2 may be con- 
sidered indefinitely small under ordinary conditions of 



Art. 6.] SHEARING STRESSES AND STRAINS. 15 

stress in structural material in order to have the stress 
uniformly distributed on the four faces. Whatever may 
be the condition of stress at any point in the interior of a 
piece of material, the stresses acting upon the four faces 
of the rectangular prism, when all stress is parallel 
to one plane, may be resolved into normal and tangential 
components. The normal components will act opposite 
to each other producing no moments about any point, 
but the tangential components will produce precisely 
the moments shown in Figs, i and 2. The equilibrium, 
of the indefinitely small prism invariably requires there- 
fore the action of two pairs of shears of equal intensity, 
as established above. 

The complete distortion of the rectangular prism 
A BCD may be considered as produced first by the sliding 
over each other of the indefinitely thin vertical sections 
parallel to BC, so as to produce the oblique prism AB"C2D^ 
Fig. I, then by the subsequent sliding over each other of 
the indefinitely thin horizontal sections parallel to DC, 
so as to produce the oblique prism AB"C"D' . This last 
movement of the horizontal slices will bring the line AD 
into the position oi AD' , then swinging the latter line about 
A to the original position AD, the completely distorted 
prism will take the form A BCD. 

B'B", Fig. I, is the characteristic shearing strain pro- 
duced by the vertical shearing stress whose intensity is 
5 acting in the planes parallel to BC. DD' is the character- 
istic shearing strain produced by the action of the hori- 
zontal shearing intensity s' in sliding the thin horizontal 
slices over each other. These detrusive movements are 
so small that B'B" may be considered at right angles to 
AB and DD' at right angles to AD. The total detrusive 
strain B'B is the sum of B'B" , due to the vertical shear, 
and B"B due to the horizontal shear, and B'B" =B"B, 



1 6 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

if AB' =AD. The total shearing strain per unit of length 
of AB will therefore be, 

B'B ^ B'B"+B''B 

AB ' AB ^^^ 



This is the expression for the characteristic resultant 
shearing strain and it is seen to be measured at right angles 
to the original face AB' , i.e., it is a small arc measurement 
in radians. It is important to remember that this total 
detrusive strain due to shear is the sum of two equal 
effects, one of horizontal shear and the other of vertical 
shear, i.e., of the two shears on planes at right angles to 
each other. 

lib = 1 and if AB'C'D, Fig. 2, now be considered square 
so that AB=BC, then will the tension T acting perpen- 
dicular to the plane BD be equal to the sum of the com- 
ponents of the shear s=s', on the planes BC and DC, 
normal to the diagonal plane BD. Since the angle BCA 

is 45° and its cosine —/=-, the following equation at once 

V 2 

results: 

r=25 cos 45° =5\/2 (3) 

Similarly the compression on the diagonal plane AC is: 

C=-s\/'2, (4) 

As the area of each diagonal plane section AC and BD 
is a/2, the intensity of the tension T and compression C 
on the planes AC and BD respectively will be: 

-r= — 7-=^ (5) 

V2 V2 



Art. 6.] SHEARING STRESSES AND STRAINS. 17 

Hence it is seen that when the stress it any point is 
wholly shear on two planes at ri'ght angles to each other 
and perpendicular to the plane to which the shearing 
stress is parallel, the stress on two planes at right angles 
to each other and making angles of 45° with the two 
planes on which the shears act, will be wholly tension on 
one and compression on the other, and both will have the 
same intensity as the two shears. 

Inasmuch as the prism whose section is shown in Fig. 
2 is subjected to a normal stress of tension in the direction 
of ^C and an equal normal stress of compression in the 
direction BLl, it is obvious that there will be no change 
in volume due to those stresses, since the change in inten- 
sity caused by one stress will be exactly, neutralized by the 
other. Again the sliding over each other of the thin 
slices of the material will not change its density or volume, 
although a change of shape is produced. Hence it is to 
be carefully observed that shearing stresses produce no 
change of volume, but change of shape only. 

If cf) is the angle B'AB =C'DC, then in general, the 
resultant shearing strain B'B =C'C =AB' (t)=AB' sin </> 
= AB' tan 0, -since the angle is exceedingly small. If 
AB=BC = 1, B'B = 4) =sm 0=tan 0. 

In Fig. 2 if the total detrusive strain CC be projected 
on the diagonal AC the change CCi in length of that 
diagonal will result. As the angle C^CCi is 45°, the 

change of length CCi will be -7=, and the strain per unit 

V2 

of length of the diagonal will be, 

-7^-7--- (6) 

V2V2 2 

It is clear that the diagonal BD will be shortened by 
the same amount. Indeed Eq. 6 shows the tensile strain 



i8 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

in the diagonal AC, while the same value with a minus 
sign would show the coriipressive strain for the diagonal 
BD. If the diagonal AC were subjected to the tensile 

intensity 5 only the strain per unit of length would be — . 

If G is the modulus of elasticity for shearing, the 
intensity of shearing stress may be written, 

s=s'=G<i>. ....... (7) 

Inasmuch as the total detrusive strain per linear 
unit is the sum of the equal effects of the shears on the 
two faces of the prism, it would be more rational to call 

— the detrusive strain per linear unit for the shear on one 
2 

face of the prism. This would make the modulus G 
of elasticity for shearing double the value usually employed, 
but it would represent accurately the rigidity of the material, 
since one half of the total shearing strain 0, Fig. 2, is pro- 
duced by a rotation of the prism as a whole. In other 
words the total strain is the sum of two separate but equal 
strains. This doubling of the value of G would obviously 
change no results of computation for practical purposes 
since the strain would be halved. It is interesting 
to observe in connection with this feature of the matter 
that the shearing rigidity of the material in this case, would 
become the same as the apparent rigidity in tension or 
compression. - 

Art. 7. — Relation between Moduli of Elasticity and Rate of 
Change of Volume. 

The preceding analyses yield some simple relations 
between the moduli of elasticity for tension, compression 



Art. 7.] RELATION BETWEEN MODULI OF ELASTICITY. 19 

and shearing and the rate of change of volume z',* i.e., the 
change of unit volume for unit intensity of stress. 

In Fig. 2 of the preceding Art. CC shows the total shear- 
ing strain 0, and the elongation or strain CCil =— ^j 

of the diagonal AC. It has also been shown that the inten- 
sity of tension on BD or compression on AC is the same 
as the shear s=s'. Remembering that the compression 

s on AC will produce a unit positive lateral strain r — 

K 

parallel to AC, the two equal values of the unit strain of 

the diagonal AC may be written, 



5 , 5 ' 
— Vf — 
2 2G \E Ej 



Hence, 

E El 



2(1 +r) 2(1 +ri)' 



(i) 



If the modulus of elasticity for compression, Ei, should 
be different from that for tension it4s evident that the third 
member of Eq. i would be required. 

If the value of r is J or J then will, 

G=iE or E (2) 

The relation between v and E can readily be written 
by considering a cube (indefinitely small if necessary) 
subjected to uniformly distributed tensile stress of inten- 
sity p normal to each of its six faces. Each edge of the 
cube, assumed to be of unit length, will be lengthened by 

the normal stress parallel to it to the extent ^, and it 

E 

* The reciprocal of what is sometimes called the volume or bulk modulus. 



20 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. 1. 

will be decreased in length r ^ by each of the two normal 

/ / 

stresses p acting at right angles to it. 

The resultant change in length of each edge will then be, 

|(i-2r). 

Hence the change of unit volume in terms of the unit 
rate v wdll be, 

pv=3^(i-2r). 
.-. E = ^-^^^^. ...... (3) 



If V be any volume, the total change of volume will 
be pvV. 

The equation preceding Eq. (3) show^s that the unit 
rate of change of volume v is simply the sum of the three 

linear rates of change of the edges of the cube, since — =^ 

K 

is the change of length of each edge of the cube for each 

/ 1 — 2T\ 

unit of ^, i.e., ^ I — - — 1 is the change in length of each such 

edge under the action of the intensity of stress p. If the 
intensity of stress parallel to each edge of the cube should 
be different from the others the preceding analysis shows 
that the rate of variation of volume w^ould still be the sum 
of the three coordinate linear rates of variation. 
By the aid of Eq. (i), 



Therefore : 



E = 2G{i-\-r)=^^^^-^ (4) 



S —2Gv , . 



Art. 8.] ALL STRESSES PARALLEL TO ONE PLANE. 

Finally, placing r from Eq. (5) in Eq. (3), 

Z7_ *9^ 



3-\-Gv' 



21 



(6) 



These simple relations will enable the various moduli 
to be determined with the least possible amount of experi- 
mental work. 



Art. 8. — All Stresses Parallel to One Plane — Resultant Stress 
on any Plane Normal to the Plane of Action of the Stresses. 

In Fig. I let XOY be the plane parallel to which all 
stresses act. Then OX and OY being any rectangular 
coordinate directions, consider the two planes OA and OB 




Fig. I. 

normal to each other and at right angles to the plane 
XOY and let the width of each of those planes at right angles 
to XOY be unity. 

Again let it be supposed that the normal stress on the 
plane AO has the intensity py and that the intensity of 
the tangential or shearing stress on the same plane is pyx- 
Similarly let it be supposed that the intensity of the normal 
stress on the plane OB is px, the intensity of the tangential 



22 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

or shearing stress being pxy. It is known from the prin- 
ciples already established in the preceding articles that the 
two intensities of shear pyx and p^j, are equal to each other- 
The problem is to determine the intensity and direction 
of the resultant stress on any plane AB, taken at right 
angles to XOY. In general the resultant stress CD will 
make the angle with the normal CF to the plane AB, 
i.e., the resultant stress will have the obliquity </>. 

The direction of the plane AB wHll be fixed by the angle 
which its normal CF makes with the axis OX. In order 
that the stresses on the three planes in question may be 
taken as uniformly distributed let it be assumed that 
OA =dx and OB =dy. Then will 

AB = dy sec a = dx cosec a (i) 

If p is the intensity of the uniformly distributed result- 
ant stress on AB, then the equilibrium of the indefinitely 
small triangular prism OAB requires that the two following 
equations, representing the sums of all the forces acting 
upon it in the tw^o coordinate directions, shall be true. 

pxdy-\-pxvdx=p cos (a + 0). dy sec a . (2) 

pydx+pxydy =p sin {a-\-4>). dy sec a . (3) 

Fig. I shows that dy tdm a=dx. Hence Eqs. (2) and 
(3) become Eqs. (4) and (5), respectively: 

px cot a-\-pxv=p COS {a-\-4>) COSeC a . . (4) 
py-\-pxv <^0t a=p sin {a -\-4>) COSec a. . . (5) 

It is sometimes convenient to express the normal and 
tangential components of the resultant intensity p in terms 
of the known intensities px, py and pxy. If in Fig. i the 
stresses on the faces OA and OB be resolved into compo- 



Art. 8.] ALL STRESSES PARALLEL TO ONE PLANE. 23 

nents normal and parallel to the plane AB the sum of the 
normal components will be equal to the normal stress on 
AB while the sum of the parallel components will be equal 
to the tangential or shearing stress on AB. This pro- 
cedure will give, 

pf4^ sin a-\-pyxdx cos a+pxdy cos a-^pxydy sin a 

= pdy sec a cos </>. 

pydx cos a—pyxdx sin a—pxdy sin a+pxydy cos a 

= pdx cosec a. sin 0. 

Using the values already given for dy and AB the fol- 
lowing expressions for the normal and tangential compo- 
nents of p (p cos (f) and p sin <t>) will result: 

py sin^ oi+px cos^ (x + 2pxy sin a cos a =;^ cos . (4a) 
(py—px) sin a cos a-{-pxy(cos^ a—sin^ a) =p sin 0. (5a) 

These two equations will be used in establishing the 
ellipse of stress in the next Art. 

If the stress ^ is a principal stress its obliquity </>, i.e., 

the angle between its direction and the normal to the 

plane on which it acts, will be zero. If 0=c Eqs. (4) 
and (5) become, 

p—px=pxv tan a, .... (6) 

p-py=pxv cot a (7) 

Subtracting Eq. 6 from Eq. 7, 



cot a — tan a 



2 


Px-py 


tan 2a 


Pxy 


2pxy 





tan 2a = — ^— ^ fg) 



24 ELASTICITY IN AMORPHOUS SOLID BODIES [Ch. I. 

If the angle ai satisfies this equation, then will ai+go"^ 
also satisfy it. Hence, there will always be two prin- 
cipal planes at right angles to each other on each of which 
a normal stress only acts, i.e., there is no shearing stress 
on either principal plane. 

Eq. 8 will at once locate, by the two values of a, the 
two principal planes, w^hile the same two values of a intro- 
duced into either Eq. 6 or Eq. 7 will give the two intensi- 
ties of principal stresses to be called pi and p2, it being 
supposed that the normal and shearing stresses on the 
planes OA and OB are completely known. 

The two principal stresses can however readily be 
found without computing the angle a. Multiplying Eq. 
7 by Eq. 6, 

P^-P{px+Pv)=pxy^-P.pu. 

The solution of this quadratic equation gives, 



2 



(9) 



The two roots of this equation will give the two prin- 
cipal intensities at any point in terms of the known inten- 
sities px, py and pxy. 

The two stress intensities px and py have been taken 
of the same kind, tension or compression, and considered 
positive. If one, as py, be considered compression or 
negative, its sign would be changed in the preceding 
equations, but there would be no other change. 

Sum of Normal Components. 

If any other plane be taken at right angles to XOY, 
Fig. I, and at right angles to the plane whose trace is AB, 
the preceding equations are made applicable to it by writing 



Art. 8.] ALL STRESSES PARALLEL TO ONE PLANE. 25 

()o°-{-a: for a in Eqs. (4a) and (5a), since the new plane 
is at right angles to that whose trace is AB. 

Then in Eqs. (4a) and (5a) there must be written, 

For sin a, sin (90+a) =cos a. 

For cos a, cos (qo+q;) = —sin a. 

Hence by Eq. (4a), writing p' and </>' for p and <^; 

py COS^ a-\-pjc sin^ a — 2pxy sin a COS a=p' COS </)'. 

Then by adding this equation to Eq. (4a) ; 

px-\-pv=p cos 4)+ p' cos <t)\ . . . (10) 

This equation shows that on any two planes at right 
angles to each other the sum of the normal intensities will 
be constant and equal to px+py. Furthermore, inasmuch 
as there is no shear on the principal planes, i.e., the stress 
is wholly normal, it is thus seen that the sum of the normal 
intensities on any two planes at right angles to each 
other is alwa3^s equal to the sum of the two principal 
intensities. 

If the above values of sin a and cos a are written in 
Eq. (5a), the following equation will result: 

(py — px) sin a cos a-\-pjy(cos^ a—sin^ a) = —p' sin <^'. 

This equation is identical with Eq. (5a), except that 
the sign of the second member is changed. This result 
simply shows what is already known that the intensities 
of the shears on planes at right angles to each other are 
equal. The change of sign indicates the direction only of 
the shear. 

In all the usual cases of stress arising in the subject of 
Resistance of Materials the internal stresses produced by 



26 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

external loading may be considered parallel to one plane. 
The preceding investigation shows that without considering 
the elastic properties of the material there are two equations 
of condition (Eqs. 4 and 5) from which the two rectangular 
components of the resultant stress p (or intensity p and 
obliquity 0) may be found. If the general case of internal 
stress be taken in which stresses may act in three rectangu- 
lar coordinate directions, there obviously will be three 
equations of condition from which the three rectangular 
components of the resultant stress on any plane may be 
found. 

The triangle OAB may be considered the side of a 
wedge whose edge is at 0. The two faces OB and OA are 
acted upon by the stresses indicated and their resultant 
holds in equilibrium the stress p on the head AB oi the 
wedge. The surface AB may be considered a part of the 
exterior surface of a body acted upon by the stress whose 
int?nsity is p, while the faces OA and OB are interior 
surfaces of the body acted upon by the internal stresses 
shown. 



Art. 9. — The Ellipse of Stress — Greatest Intensity of Shearing 
Stress — Equivalence of Pure Shear to Two Principal Stresses 
of Opposite Kinds but Equal Intensities — Greatest Obliquity 
of Resultant Stress on any Plane. 

The analysis of the preceding article makes it compara- 
tively easy to express the relation between the stress on 
any plane whatever at right angles to the plane parallel 
to which the principal stresses act and those principal 
stresses, all stresses still acting parallel to one plane. In 
Fig. I let OX and OF be taken in the direction of the 
principal stresses, OA representing the intensity pi of the 
principal stress at on the plane OD, w^hile OB represents 



Art. 9.] THE ELLIPSE OF STRESS. 27 

the intensity of the principal stress p2, acting at O on the 
plane OC. OCD represents an indefinitely small triangular 
prism whose face CD normal to XOY makes the angle jS 
with the principal plane OD. The intensity of the re- 
sultant stress on any plane CD is represented by p, whose 
obliquity is 0, the normal N to the plane CD making the 
angle ^ with the axis OX. The resultant intensity p may 
at once be written by the aid of Eqs. 4 and 5 or 4a and 
5a of the preceding article if the principal intensities pi 
and p2 be written in the place of px and py, respectively, 
in those equations while pxy is made equal to zero. This 
procedure with Eqs. (4a) and (5a) will give the following 
Eqs. (i) and (2). 

^2 sin^ /5+:^i cos^ /3 =;/? cos 0, . . . (i) 

{p2 — pi) sin jS cos 13 =— — — sin 2(3 =p sin 0. . (2) 

2 

Squaring each of those equations and adding the results : 

p2^ sin^ 13+ pi^ cos^ ^=p^. . . . (3)* 

This is the equation of an ellipse with the origin of 
coordinates at the centre, the rectangular coordinates being 
p2 sin jS and pi cos 13. Fig. i shows the ellipse of stress, 
the intensities of the principal stresses being represented 
by the semi axes of the ellipse. 

0B=p2 and OA=pi. 

In this Fig. p2 represents the intensity of the principal 
stress on the plane OC, while pi is the intensity on the 
principal plane OD. The intensity p on any plane as CD 

* Precisely the same result will be obtained by making ??2-2/ = o in eqs. 
4 and 5 of the preceding Art. and then squaring and adding them. 



28 ELASTICITY OF AMORPHOUS SOLID BODIES. [Ch. I. 

perpendicular to XOY and whose normal ON makes the 
angle /8 with OX is represented in Fig. 2 by OH, the curve 
AHB being an ellipse. Let the partial circles shown be 
described by the radii OB and OA. Then if OCD be con- 
sidered indefinitely small the normal ON, and the line 
OH representing the intensity of the resultant stress on 
the plane CD, will both pass through the origin 0. Then 
OG will represent p2 and 0K=p2 sin ,8. The same con- 
struction shows that HK=pi cos ^ since OJ =pi. The 
square of OH = p will then obviously be equal to the square 
of HK added to the square of OK, an expression identical 
with Eq. (3). 

Any radius vector of the ellipse therefore represents 
the intensity of a resultant stress on a plane whose normal 
makes the angle /3 with the axis of X. The obliquity of 
the resultant stress in question is represented by the angle 0. 

The two principal stresses have been taken of the same 
kind in finding the ellipse of stress, but the results are es- 
sentially the same if the principal stresses are of opposite 
kind. If for example, p2 were negative while pi remains 
positive p2= OB would be laid off in Fig. i to the left 
of instead of laying it off to the right of the same point. 
Similarly if the sign oi pi should be considered negative 
that intensity would be laid off downward from to A^ 
instead of upward to A. 

If the two intensities of principal stresses pi and p2 
are equal to each other and of the same kind Eq. 3 becomes 

Pl=p2=p. 

Under the same conditions Eq. (2) shows that the 
shearing intensity is zero, whatever value the angle (3 
may have, since in such a case pi—p2=o. Hence all 
stresses are principal stresses and of equal intensity. This 
condition of stress is the same as that which holds in a 
perfect fluid. 



Art. 9 



THE ELLIPSE OF STRESS. 



29 



An examination of the ellipse of stress as given in 
Fig. I shows that the intensity of one principal stress is 
greater than that of any other stress at the point for which 
the ellipse is drawn, while the intensity of the other prin- 
cipal stress is the least of all the intensities at the same 
point, since the semi-major and semi-minor axes of the 
ellipse are the greatest and least, respectively, of all the 
semi-diameters. If therefore in the design or construc- 
tion of any machine or structure the principal stress at 




any point is provided for by the use of a proper working 
stress, no further provision for the direct stresses of ten- 
sion and compression will be needed. If there may be 
either a reversal of stress or rapid repetition of stresses the 
intensity of working stress must ^e prescribed under a 
proper recognition of those conditions. Similarly provi- 
sion must be made for the greatest shearing stress at the 
point under consideration. 



Greatest Intensity of Shearing Stress. 

The intensity of shear on any plane CD at the point 
0, Fig. I, is ^ sin (^ as given by Eq. 2. Its greatest value 



30 ELASTICITY IN AMORPHOUS SOLID BODIES. fCh. I. 

and the plane on which it acts are readily determined by- 
differentiating that equation: 

^^^^-^^=-{p2-pi)(cos^^-sm^^)=o = {p2-pi) cos 2/3. 

Hence, 

coS|S=sini3; or, /3=45° (4) 

As sin 45° =cos 45° =— T=, the greatest intensity of 

V2 

shear at any point, as O, Fig. i, is found by substituting 
/3 =45° in the second member of Eq. (2) : 

p.^^^^^. ...... (s) 

2 

The planes of greatest shear, therefore, are at the angle 
of 4S° f^om each of the two principal planes, and the greatest 
intensity of shear is half the difference of the principal 
intensities, both of the latter being of the same kind. 

As /3=45° the resultant intensity of stress on the plane 
of greatest shear will be, by Eq. (3), 



If p2 = dtpl ; p = zLp2 = zbpl. 



P^=il±£^ .-. P^^Ji^±£^. . . (sa) 



Equivalence of Pure Shear to Two Principal Stresses of 
Opposite Kinds but Equal Intensities. 

If the principal stresses are of opposite kinds, i.e., if 
pi is negative w^hile p2 is positive, then by Eq. (5) the 
greatest shear becomes : 

p,-^-i^^ (6) 



Art. 9.] EQUIVALENCE OF PURE SHEAR. 31 

The greatest intensity of shear is half the sum of the prin- 
cipal intensities. 

Obviously the planes of greatest shear remain as estab- 
lished by Eq. (4). 

If the principal stresses of opposite kinds have the same 
intensities Eq. (6) shows that: 

pt=p2=pi (7) 

Hence the intensity of the greatest shear is the same 
as that of the principal stresses of opposite kinds. It is 
therefore sometimes stated that a pair of normal stresses 
of opposite kinds and equal intensities on two planes at 
right angles to each other are equivalent to two pure 
shears of the same intensity as the normal stresses on 
planes at right angles to each other, but at 45° with the 
planes on which the normal stresses act, all planes under 
consideration being perpendicular to one plane. This 
simple condition of stress exists in both flexure or bending 
and torsion, and some important results are based on it. 

Greatest Obliquity of Resultant Stress on any Plane. 
If Eq. (2) be divided by Eq. i: 

4. , ( s^ ^\ sin j3 cos/3 zon 

tan = (^2-i?i) -Or.. ^— • • • (8) 

:p2sm2 /3+^i cos2/9 ^' 

It is desired to find that value of ^ which will make <^ 
(or tan 0) a maximum. By differentiating Eq. (8) and 

placing — ^ — . =0, there will result, 

(2/3 

(cos^ /8 -sin2 /3)(^2 sin^ ^^px cos^ ^) 

= 2 sin2 /? cos2 I3(p2 -pi). . . . (9) 



32 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

Remembering that cos^ i3— sin^ i3=cos 2/3 and that 
2 sin /? cos i3=sin 2/3, Eq. (9) will become Eq. (10); 

cos 2/3(^2 sin^ 13+pi cos^ /3) =sin ?.(3 sin j0 cos ^{p2—p\). (10) 

Calling the normal component of the intensity p, i.e., 
p cos (t)=pn and the tangential or shearing component 
^ sin <l)=pt, those values taken from Eqs. (i) and (2) 
placed in Eq. (10) will give, 

cos 2^pn =sin 2^pt. 
p 

Hence, tan 2/3=— = cot = tan (90° — 0). . . (11) 

pt 

And, ^=45°-- (12) 

2 

Eq. (12) gives the relation between (3 and when the 
obliquity <^ is the greatest possible. 
By the aid of Eq. (12), 

sin /3 cos ^ =^ sin 2(3=^ cos 0. 
Then as, sin2(45°-^^ =-(i -sin 0), 



and 0052/45° — j =-(i +sin 0), 

Eq. (8) gives. 



, sm . ^ s cos 

tan0= -={p2-pi)~ 



cos p2{i —sin 0) +^i(i +sin 0)* 
Hence, 

pi I — sin . . ^ p2—pi f V 

— = ^ — ^, and, sm0=^^- ^—. . . (i^) 

p2 i+sm :/?2+^i ^ 



Art. lo.] ELLIPSE OF STRESS. :^s 

The relation shown in the first of Eqs. (13) is used in 
the theory of earth pressure. The second of Eqs. (13) 
gives the value of the greatest obliquity in terms of the 
known principal intensities pi and p2. 

The "angle jS locating the plane on which the obliquity 
is greatest may also be expressed in terms of pi and p2. 

Using Eqs. (12) and (13), 

pi _i —sin _ I —cos 2/3 sin^ jS 



p2 I +sin I+COS2/5 cos 



The intensity, p\ of this stress of greatest obliquity 

is, by Eq. (3), since by Eq. (14) sin^ (3= — ^ — and 

pi +^2 

cos^ jS = — , 

pi+p2 

pi=\/pip2 (15) 



Art. 10. — Ellipse of Stress and Resulting Formulae for the Special 
Case of Zero Intensity of One of the Known Direct Stresses. 

If in the second preceding article it be supposed that 
the intensity of one of the direct stresses as px is zero 
while the other intensity py and the two shearing inten- 
sities pxy=pvx have known values, the formulas will be 
correspondingly simplified. This is the condition of stress 
in a bent beam as will be seen later on. The intensity 
of direct stress py is what is ordinarily called the fibre 
stress at any point in the beam and this intensity varies 
directly as the distance from an intermediate plane (before 
flexure) in the beam called the neutral surface. The 
plane OF of Fig. i, representing part of a beam, is su'p- 



34 



ELASTICITY IN AMORPHOUS SOLID BODIES. ICh. I. 



posed to be a horizontal plane coincident with or parallel 
to the neutral surface of the bent beam at any point, while 
the plane whose trace is OX is the plane of vertical (normal) 
transverse section of the beam at any point. Both the 
direct intensity py and the intensity of shear ^^y are 
readily determined from the known conditions of loading 
and flexure. The analysis of this condition of stress 
therefore is of much practical importance in connection 
with the design or other treatment of beams subjected to 
transverse bending. 



CL 



m 




i 



Fig. I. 



If the stress ^ in Eq. (4a) of Art. 8 is a principal 
stress and if the intensity px = o, the principal intensity 
p will become, 

p =py sin^ a + 2pxy sin a cos a. ... (l) 

Or, Eq. (9) of the same Art. will give for the two prin- 
cipal intensities, 

p=hPv±'^P^v'+lPv' (2) 

Also Eq. (8) of the same Art. will give, 

tan2c.= -^ ...... (3) 



Art. 10.] ELLIPSE OF STRESS. 35 

If the point 0, Fig. i, is in the neutral surface of the 
bent beam py=o\ and, hence, 

tan 2q: = — oo , or, 20: = ±90°. ... (4) 

Thieref ore , a = d= 4 5 ° . 

If the stress py is negative or compression, a= \ ^^q. 

[ +135 
The direct fiber stresses in a bent beam are tensile on one 
side of the neutral surface and compressive on the other. 

As in this special case q;=— 45°, sin a=— cos a= ^ 

V2 

and the intensity p of the principal stress becomes by the 
aid of Eq. (i), since py=o. 

p=-p.y. ...... (5) 

It has already been seen that a and 90°+ a will satisfy 
Eq. (3); but 90° +q; =90°— 45° =45°. Hence placing 
a = +45°inEq. (i), 

P = +Pxy (50^) 

Therefore at O, Fig. i, where there is no direct stress (but 
shear only) on the two planes OX and OY the principal 
stresses are of equal intensities, but of opposite kinds and 
they act on planes making angles of 45° with the planes 
OX and OY. This is the same condition shown by Eq. 
(7) of the preceding Art. 

Again at the exterior surface of the beam p has its 
greatest value and the shearing intensity pyx=o. Eq. (2) 
then gives, 

2«=o or 180°. 

Hence, a=o or 90° (6) 



36 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I- 

Eq. I gives p=o for the principal stress corresponding to 
a=o; and, for a = 90°, 

P=Pv (7) 

There is therefore only one principal stress py, the 
fiber stress acting on the normal section of the beam for 
which a =90°. 

For intermediate points of the beam between the 
neutral surface and the exterior surface the principal 
stresses will have varying values between pxy and py as 
shown by Eqs. (5) and (7) with planes of action located 
by values of a between ±45° and +90°. 

A graphical representation of this condition of stress 
for a bent beam may be found in Art. 34* 

Art. 1 1. ^General Condition of Stress^Ellipsoid of Stress. 

The conditions of stress in structural material as found 
within the experience of engineers seldom include more 
than the action of stresses parallel to one plane. There 
may, however, be occasional cases in which an elementary 
consideration of stresses acting in any direction whatever 
becomes necessary or at least helpful. In this Article 
therefore only the most elementary results of the action 
of such stresses will be treated, including the ellipsoid 
of stress. 

In the preceding articles both the determination and 
the application to a number of useful cases of the ellipse 
of stress have been made. That ellipse is simply a special 
case of the more general ellipsoid of stress. In other 
words, if the action of stresses in space, i.e., on three 
rectangular coordinate planes be considered it will be found 
that there will be three such planes at any point on which 
there will be no shear and which therefore are called prin- 



Art. 



GENERAL CONDITION OF STRESS. 



37 



cipal planes, the resultant normal stresses being called 
the principal stresses at that point. The semi-diameter 
of the ellipsoid of stress drawn with its center at the point 
under consideration will be the intensity of stress in that 
direction, acting upon a plane whose position may be 
determined. For this elementary treatment let the three 
rectangular coordinate planes in Fig. i be drawn. 




Fig. I. 



In that Fig. the normal stresses on the planes XOY, 
YOZ, and ZOX have the intensities pz, px and py, respect- 
ively. The intensities of the shearing stresses on the 
planes XOY and XOZ, parallel to YOZ, are pzv=pyz\ 
and those on the planes XOY and YOZ, parallel to ZOX, 
are pzx = pxz ; and finally those on the planes YOZ, and 
XOZ, parallel to XOY, are pyx=pxy. If these normal and 
shearing or tangential stresses on the three faces, AOB, 
BOC, and AOC of the elementary tetrahedron A ECO 



38 ELASTICITY IN AMORPHOUS SOLID BODIES. . [Ch. I. 

are given, it is required to find the resultant intensity of 
stress on the plane surface ABC, the base of the tetra- 
hedron, and its obliquity. It may be considered that 
AO=dx, BO=dy, and CO=dz. It will simplify the result- 
ing equations if there be written for the areas of the 
faces of the elementary tetrahedron; 



dxdy , dydz . 

a= -\ 0=^^ — ; and c = 



dxdz 



Also if area ABC = J, and if the angles which the normal 
N to the face ABC makes with axes of X, Y, and Z, 
respectively, are a, /3, and ;-, there may be written: 

i =a sec ;' = 6 sec a =c sec i3. . . . (i) 

The tetrahedron is held in equilibrium by the normal 
and shearing stresses on the faces a, b, and c and the 
resultant stress whose intensity is q on ABC = J. The 
components of that resultant parallel to the axes of X, 
Y, and Z wdiose intensities are Qx, q^, and qz are respect- 
ively equal and opposite to the corresponding axial sums 
of stresses as shown by the following equations : 

pzh+pzxa+pyxC=qxA, ' (2) 

puC-\-pzya-{-pxyb=qyJ, (3) 

pza+pxzh+pyzC=qzA, (4) 

As these are rectangular components, if their squares 
are added the sum will be equal to q^I^. If both sides 
of the resulting equation be divided by J^, remembering 
that 

--=cos2 7'; --=cos2a; — = cos2j8; 



Art. II.] GENERAL CONDITION OF STRESS. ' 39 

ah be ^, ^ ac _ 

— - = cosacosr; — 7 = COS a COS iS ; and -- = cos /S cos r ; 

and that 

COS^ a+COS^ /3+COs2 r = 1; • • • • (s) 

there will be found : 

p/ cos^ cx-\-py^ cos^ (S+p-^ cos^ y-\-2 cos a COS r{pxpzx-\- 

Pxvpzy+Pzpxz) +2 COS a COS ^{pxpyx+pxzpyz^-pypxy) + 

2 COS /3 COS riPzPvz+Pzxpxy+pypzy) +P^^xz{l " COS^ /S) + 
^\:,(l-COsV)+.^^2^(l-COs2a) =g2 (6) 

The square root of the first member of eq. (6) will 

give the desired value of the intensity q on any given plane. 

If both members of eqs. (2), (3), and (4) be divided by J : 

px cos a +pzx cos r+ pyx cos (3= qx, , , . (7) 
pjj COS ^+pzy COS r+Pxy COS a=qy, . . . (8) 
pz COS r +Pxz COS a +pyz COS /3 = g^. . . . (9) ' 

If p be the angle between the axis of X and the direc- 
tion of q, then will 

cos pi =— (10) 

Eqs. (8) and (9) give similar values of the cosines of 
the a.ngles between the direction of q and the axes of Y 
and Z, thus fixing the direction of q. 

Using the values of qx, qy, and qz as given in eqs. (7), 
(8), and (9), the component of q normal to its plane of 
action {A BC = J) will be : 

qn=qx cos a +qy cos (3 -\-qz cos y. . . . Cii) 



40 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. T. 

Hence the obliquity of ^ can at once be determined 
by the equation 

cos 0=— (12) 

The triangular face XYZ of the tetrahedron Fig. i 
may be considered a part of the exterior surface of a body 
on which acts the stress whose intensity is q. The three 
rectangular coordinates faces XOY, YOZ, and ZOX are 
then to be taken as interior surfaces of the body on which 
act the internal stresses indicated. The stress on the 
external face XYZ must be in equilibrium with the stresses 
acting on the three interior rectangular coordinate faces 
of the tetrahedron. 

Principal Stresses and Ellipsoid of Stress. 

The preceding equations are general and relate to 
stresses on any planes whatever. If, however, the stress 
q is normal to its plane of action it is a principal stress. 
In that case the obliquity is zero and there is no shear. 
Hence, 

qx=q cos a] qy=q cos (3; qz=q cosy. . . (13) 

• 

Substituting these values in the second members of 
eqs. (7), (8), and (9), and then eliminating cos a, cos /3, 
and cos r from the three resulting equations, the follow- 
ing equation of the third degree will be found : 

q^ - (px +py +Pz)q^ + (Pxpy +Pxpz +pypz- P^xy - p^x -p\y)q 

-\-pxp'^zy-^PvP'^zx-\-pzp-xy-pxpypz-2pxypzxpyz=0. . . (14) 

Or, indicating the coefficients of q and the part of this 



Art II ] PRINCIPAL STRESSES AND ELLIPSOID OF STRESS. 41 

equation independent of that quantity by A, B, and C, 
respectively : 

q^-Aq^+Bq-C = o (15)* 

The three roots of this cubic equation are the inten- 
sities of the three principal stresses, and the equation 
shows that at every point three such principal stresses 
exist, each normal to its plane of action on which there is 
no shear. 

If in, eq. (6) the coordinate axes of X, Y, and Z be 
taken as the principal axes so that the intensities px, py, 
and pz become the principal intensities gi, qo, and ^3, 
then will. pxy=pyz=px2 = o, and q will be the intensity 
of stress in any direction on a plane whose normal makes 
the angles a, ^, and 7 with the coordinate axes, i.e., with 
gi, ^2, and gs. Hence 



q = '\/qi^ cos^ a-\-q2^ cos^ iS+^s^ cos^ ;- . 



(16) 



Again, if g,, qy, and qz are the rectangular components 
of q, Eqs. (7), (8), and (9), will, under the same conditions, 
give: 

qi cos a=qx', q2 COS ^=qy; and qzcosy=q2. 

Then, squaring and adding: 

^4+s4+si^=, (17) 

<?r q2^ qs^ 

* Rankine observed in his Applied JMechanics that if qi, q-j, and §3 are 
the roots of a cubic equation, then : 

{q-qi){q-q^{q-qz) =q'^ -q-{(li+q.-i-^r<lz) +q{qiq-i+q-2q^+qxqz) -qiqm=o. 

This equation shows that the quantities A, B, and C remain the same 
whatever may be the directions of the three rectangular axes at a given 
point. Hence, by using A it is seen that qi.+qs-\-q3=px+p!/-\-pz, i.e., the 
sum of the normal components of the intensities of stress on any three 
rectangular coordinate planes is constant and equal to the sum of the 
intensities of the three principal stresses. 



42 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

By dividing this equation through by q^ it may be 
written in terms of the angles between q and the coor- 
dinate axes (eq. i8) and the reciprocal of q^. 

Eq. (17) is the usual form of the equation of an ellip- 
soid with the origin of coordinates at its center in which 
qi, q2, and qs are the semi axes and qx, qy and qz are the 
coordinates of any point in the surface. 

The intensity of stress q in any direction, represented 
by the semi-diameter of the ellipsoid in that direction, 
is given by eq. (16), and the angles between its direction 
and the coordinate axes X, F, and Z may, by the aid of 
eq. (10), be written: 



qx q\ cos a qy g2 cos 
cos pi = — = ; cos p2 = — = 

q q. q q 

qz q-i cos r 
cos P3=- = . 

q q 



(18) 



The component of q normal to its plane of action is 
given by eq. (11) : 

g„=gi COS^ q;+Q'2 cos2 /3+g3 cos^ y. . . (19) 

The cosine of the obliquity of q is therefore: 

q^ / \ 

COS =— . . . . . . . (20) 

q 

These elementary considerations are sufficient for the 
purpose of outlining to some extent at least the general 
subject of stress in any or all directions in solid bodies. 
The results may easily be developed, so as to be appHcable 
to the solution of any required problem. The equations 
(2), (3), and (4) are frequently applied to the discussion 



Art. 12.] ELLIPSE AND ELLIPSOID OF STRAIN 43 

of the action of external forces qx, qv, and qz, in connection 
with the internal stresses px, py and pz, etc., as will be 
indicated later. 

It is obvious that if all the internal stresses act parallel 
to one plane, eq. (14) and those which follow it will relate 
to the ellipse of stress, showing that the latter is a special 
case of the ellipsoid of stress. 

Art. 12. — Ellipse and Ellipsoid of Strain. 

It has been shown that the intensity of stress at any 
point in a solid homogeneous body may be represented 
by the semi-diameter of an ellipsoid in the general case 
or the semi- diameter of an ellipse in the special case of all 
stress being parallel to a plane. Inasmuch as strains are 
proportional to the corresponding stresses below the elastic 
limit, the strain of a very short but constant length of a 
solid element at any point would be represented by the 
semi-diameter of an ellipsoid or ellipse having the same 
direction as the corresponding intensity, which also might 
be represented by the same semi-diameter at a proper 
scale. It follows from these simple considerations that 
strains in all directions may be represented by ellipsoids 
and ellipses as well as stresses. While such ellipsoids and 
ellipses possess analytic interest in connection with the 
theory of elasticity in solid bodies, they are not of sufficient 
importance in the structural operations of engineering to 
require even elementary analytic treatment. 

Art. 13. — Orthogonal Stresses. 

When stresses of tension or compression at right angles 
to each other concur either in one plane or on three coor- 
dinate planes making right angles with each other, a*s in 



44 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

the cases of the ellipse and ellipsoid of stress, they are 
said to be orthogonal stresses. Such stresses produce 
partially independent strains in the directions in which 
they act, but the resultant stress on any one plane is a 
single stress obviously accompanied by its character- 
istic strain. This is true whether the stress is wholly 
parallel to one plane or if it acts in all directions. The 
fact that lateral and direct strains in the same directions 
may concur has induced some engineers and writers to 
attempt to provide rather arbitrarily for the supposed 
effects of orthogonal stresses and strains. 

If in the case of stress wholly parallel to one plane 
px and py represent the intensities of the principal stresses, 
as in Art. 7, the unit strain parallel to the axis of x 
will be, 

Similarly the unit strain in the direction of y will be, 

7 ^^^ px ■ . . 

^'"'E £ ^^^ 

In the preceding eqs. (i) and (2) the plus sign is to be 
used if the intensities px and py are of opposite kinds, but 
the minus sign is to be written if the two stresses are of 
the same kind, i.e., both tension or both compression. 

Two intensities of stress p'x and p'y are then assumed, 
each of which if acting separately would produce the 
strains in the two coordinate directions, respectively, 
shown by eqs. (i) and (2). These two intensities must 
have the following values: 

pxzLrpy and ppzLrpx (3) 



Art. 13:] ORTHOGONAL STRESSES. 45 

These are called " equivalent " intensities of stress, and 
it is postulated that the working intensity of stress pre- 
scribed for any member of a structure must not exceed 
the greatest of the two values given by eq. (3). 

In the special case of two principal stresses being of 
opposite kinds but of equal intensity, the greatest shear 
will be of the same intensity as the principal stresses, 
or by the aid of eq. (3) 

px=pt= -pvr (4) 

or, combining eqs. (3) and (4), 

p'x=pt-\-rpt = (i+r)pt, .... (5) 
hence, 

pt 



i+r 

In the latter case it is said that the greatest shear 
must not exceed pt in eq. (6),p'x representing the pre- 
scribed working intensity in tension or compression as 
the case may be. 

This arbitrary substitution of an intensity of stress 
corresponding to the sum of two coordinate strains, in the 
place of an actual greatest intensity of stress acting on its 
proper plane, is not supported by any substantial analytical 
or experimental basis. The maximum intensity of stress 
at any point in a piece of material subjected to loading 
may readily be determined and the position of the plane 
on which it acts may be ascertained by the methods given 
in the preceding articles, and it is difficult to imagine any 
sufficient reason for not making that actual maximum 
intensity of stress equal to the prescribed working stress 
of the same kind. The maximum intensity of stress at 
any point will of course be accompanied by the maximum 



46 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

unit strain and a proper limitation of that strain will be 
coincident with a proper limitation of the stress pro- 
ducing it. These observations are equally true whether 
the kind of stress involved be tension, compression, or 
shearing. 

The substitution of an artificial " equivalent " stress, 
therefore, in the place of the actual maximum stress at 
any point remains to be justified and will not be employed 
in this work. All the design work involving the employ- 
ment of a prescribed working stress will be based upon the 
greatest actual intensity of stress in the structural mem- 
ber under consideration. 

Again, the significance of lateral strains has been 
expressed by stating that if a straight bar of structural 
steel with square cross-section, for illustration, be sub- 
jected to a tensile stress of intensity p, the lateral strains 
will- be negative, as they decrease the lateral dimensions 
of the bar, and hence that if the ratio of lateral to direct 
strains be taken as one-fourth, then those lateral strains 
are each precisely the same as would be produced by an 
intensity of compression equal to -Ip, acting at right angles 
to the bar and on either pair of opposite sides. Hence, 
it has been said that such a bar is not only subjected to 
the axial tension, but also to a '* true internal stress which 
acts as a compression at right angles to the axis of the 
bar." It is further stated that such a bar " suffers a 
true internal compressive unit stress ... in all direc- 
tions at right angles to its length ..." 

It is still further stated that " The injury done to 
a body does not depend upon the actual stress or pres- 
sure, but upon the actual deformations produced, and the 
true stresses are those corresponding to these deforma- 
tions." 

It is difficult to imagine how the " actual " stress 



Art. 13.] ORTHOGONAL STRESSES. 47 

existing at any point in a body fails to be the " true " 
stress. If the " true " stresses are different from the 
actual they must be imaginary or at least not actual or 
real. 

It cannot be admitted that the lateral strains accom- 
panying the direct strains of a bar subjected to axial 
tension are produced by " a true internal " compression, 
for no such corresponding external compressive forces or 
pressure at right angles to the axis of the bar exist. If 
the lateral strains were due to such- compressive stress, 
the corresponding external compressive forces would per- 
form work and would make the total resilience of the bar 
two-ninths greater than the resilience due to direct tensile 
stress only, if the ratio r be taken as one-third. 

This species of confusion seems to arise at least partly 
from a failure to distinguish between molecular conditions 
below the elastic limit and those above that limit. 

If a bar is subjected to axial tension producing corre- 
sponding axial and lateral strains, in consequence of which 
the lateral dimensions of the bar decrease, it by no means 
follows that actual compression has produced that decrease. 
In fact, since the molecules have been separated to a slight 
degree axially, the transverse movement of the molecules 
may easily be conceived to take place without any 'com- 
pression whatever, and the fact that the density of the 
material is decreased by tensile stress makes that view 
reasonable, and perhaps conclusively confirms it. It should 
be remembered that all these analytic investigations re- 
late only to stresses and strains existing below the elastic 
limit. 

While it is true that experimental investigation is 
still lacking to give complete information regarding the 
effects of orthogonal stresses and strains below the elastic 
limit (as well as above it) there is lacking material evi- 



48 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

dence showing the existence of any such stress conditions 
consequent upon the existence of lateral strains as those 
to which allusions are made above, and they will not be 
recognized in the analytic work which is to follow. 

In discussing the stresses in the walls of thick cylin- 
ders in Appendix II, the bearing of these considerations 
on the formula of Clavarino will be fully set forth. 



Problems for Chapter I. 

Problem i. — A wrought iron bar 4."xi" in section is 
subjected to a tensile force of 28,000 pounds. The stretch 
for a gaged length of 20 feet was 0.12 inch. Find the 
intensity of tensile stress in the material, the modulus of 
elasticity E, and the rate of strain, i.e., the strain per 
linear inch. 

Ans. Intensity of stress = 14,000 lbs. per square inch. 
£=28,000,000 pounds per square inch. 
Rate of strain = 0.0005 inch per inch. 

Problem 2. — ^A steel eye-bar 8'^X2'' in section carries 
a total load of 128,000 pounds, under which there is a 
stretch of 0.016'' in a gaged length of 5 ft. Find the in- 
tensity of stress, rate of strain, and modulus of elas- 
ticity E. 

Problem 3. — Steel has a modulus of elasticity of 
30,000,000 pounds per square inch, and a coefficient of 
expansion of 0.0000065 per degree F. If a steel bar 2''X4'' 
in cross-section has a length of 30' o'' at a temperature 
of 40° F., find the length of the bar at 10° F. and at 110° 
F. Suppose the ends of this bar had been fastened rigidly 
at the temperature of 40° F. Find the intensity of ten- 
sile stress at 10° F.. and intensity of compressive stress at 



Art. 13.] PROBLEMS FOR CHAPTER I. 48a 

110° F., supposing the bar to be firmly held against lateral 
deflection. 

Partial Ans. Length of bar at 10° F. =29'. 99415. 

Intensity of tensile stress in bar at a 
temperature of 10° F. =5850 pounds 
per square inch. 

Problem 4. — A concrete pillar 24''X24'' in section and 
8 ft. high carries a total (compressive) load of 115,200 
pounds. If the modulus of elasticity for the concrete 
is 2,500,000 pounds per square inch, what will be the 
rate of compressive strain and the shortening, first, for 
the total height 8 ft. of pillar, and, second, for 12'', under 
the preceding load? 

Problem 5. — In Problems i and 2, if Poisson's ratio 
r (i.e., the ratio of lateral to direct strain) is 0.3, find the 
new cross-dimension of the bars and also the change in 
volume for a portion of each bar i foot long. 

Ans. for Problem i. 

d = s''.gggi6; ' b =o'\4ggSgs', 

change in volume = 0.00908 cubic inch decrease. 

Problem 6. — In Problem 3, the cross-dimensions of 
the bar, under the compressive stress, become 2''. 0001 14 
and 4^^.000228. Find the ratio r betw^een direct and 
lateral unit strains, and also the increase of volume of 3 
ft. length of the bar. 

Problem 7. — In Problems 5 and 6 find the modulus 
of elasticity, G, for shearing in terms of the direct modulus 
of elasticity E. 

Problem 8. — In Problem 2 find the total normal and 
tangential stresses and their intensities on plane sections 
making angles of 18°, 35°, and 53° with the axis of the 
piece. 

Problem 9. — In Problem 3 find the total normal and 
tangential stresses and their intensities on plane sections 



486 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

making angles of 31°, 45°, and 72° with the axis of the 
piece. 

Problem 10. — A round steel bar 3 inches in diameter 
is subjected to a tensile stress of 212,100 pounds. If the 
diameter of the bar decreases 0.00105 inch, find the 
ratio r between the direct and lateral strains, and also 
the increase of volume in a 4-ft. length of bar. Assume 
modulus of elasticity E as 30,000,000 pounds per square 
inch. 

Problem 11. — Given three planes, AO, OB, and BA, 
Art. 8, so placed that AOB =90° and AB0=a = 6o°. 

The tensile stress on OB is px=s5oo pounds per square 
inch and the tensile stress on OA, py = s^oo pounds per 
square inch. The shearing stresses on OA and OB are 
equal, i.e., pxy=pyx = i7S^ pounds per square inch. 

Find the normal and tangential components of the 
resultant intensity p, when p makes an angle </) = io°, 
below the normal to the plane AB. Also find the inten- 
sity of the principal stress on the plane AB. 

Problem 12. — In Fig. i. Art. 8, let the intensity of the 
normal tensile stress on the plane OB be 8000 pounds per 
square inch, while the intensity of normal compressive 
stress on the plane OA is 12,000 pounds per square inch, 
and let the intensities of shearing stresses on the same 
planes OB and OA be 3500 and 6500 pounds per square 
inch respectively. Find the principal stresses and the 
principal planes on which they act. Then, by means of 
the formula of Art. 9, find the greatest intensity of shear- 
ing stress on any plane at 0, and the position of that plane. 
Finally, determine the intensity of the stress of greatest 
obliquity at the point O, and the plane on which it acts, 
together with the intensity of shearing stress on that 
plane. 



CHAPTER II. 
FLEXURE. 
Art. 14.— The Common Theory of Flexure. 

A. STRAIGHT piece or bar of material is subjected to 
flexure or bending when it is acted upon by loads or forces 




Fig. I. 



at right angles to its axis, the loads and supporting forces 
taken as a whole constituting a system in equilibrium. 



49 



5° 



FLEXURE. 



[Ch. II. 



The beam shown in Fig. i may be taken to illustrate 
the general condition of flexure or bending. 

Each end of the beam is supported as shown at R and 
R\ the reactions at those points constituting the support- 
ing forces, while the weights W^ and W^, etc., constitute 
the loading. The reactions are in reality just as much 
loads on the beam as the weights carried by it, but it is 
convenient always to make the distinction between loads 
and reactions or supporting forces. 



I 



Q_ja 



mmmmmm 

I 




Fig. 2. 



An overhanging beam is shown in Fig. 2 carrying the 
weights W^ and W^, etc., one end being firmly fixed in a 
wall or other similar supporting mass. In this case the 
supporting effect of the material in which one end of the 
beam is embedded is equivalent to the couple whose 
moment is Fe. Obviously there may be many other 
different cases of bending, according to the manner of 
supporting and loading the bent piece or beam. 

In all these analyses and in all that follow, except 
when otherwise specially noted, the beams are supposed 
to be horizontal with the loads and reactions vertical, all 
external forces thus acting at right angles to the axis of 
the beam, and they are further supposed to lie all in a 
vertical plane passing through the same axis. When the 
loading acts as shown in Fig. i , it is evident that the bean] 



Art. 14.] THE COMMON THEORY OF FLEXURE. 51 

will be bent so as to become convex downward and con- 
cave upward, thus causing the upper portion of the beam 
to be in compression while the lower portion is in tension. 
Hence if any normal section of the beam as BD be con- 
sidered, in passing from B where there is compression 
to D where there is an opposite stress of tension it is clear 
that at some point, as m, there will be a zero stress, or, 
in other words, no stress at all. The horizontal line pass- 
ing through that point m of no stress, and normal to the 
vertical plane through the axis of the beam, is called the 
neutral axis of the section and its locus HX throughout . 
the entire beam is called the neutral surface. On one side 
of the neutral axis in any normal section there will be 
direct stresses of compression and on the other direct 
stresses of tension. There are two fundamental assump- 
tions in the common theory of flexure: 

First, that all plane normal sections of the beam remain 
plane after flexure or bending. 

Second, that the intensity (amount uniformly dis- 
tributed on a square unit) of either the tensile or 
compressive stress in any normal section acting 
parallel to the axis of the beam varies directly as 
the distance from the neutral axis of the section. 

In Fig. I the shaded triangles above and below m, 
having the common vertex at that point, represent the 
stresses of tension and compression induced in the normal 
section BD by the bending. 

The loads and supporting forces act normally to the 
axis of the beam upon either portion of it, as HBD, while 
the internal stresses of tension and compression in the 
section BD act parallel to that axis. If the equilibrium 
of the same portion HBD be considered, it will be seen 
that the only horizontal forces acting upon it are the in- 



52 FLEXURE. [Ch. II. 

tcrnal stresses of tension and compression shown by the 
two shaded triangles. Hence in ord^r that there nriay 
be equilibrium the sum of those stresses of tension and 
compression must be equal to zero. This latter condition 
will determine in a simple manner the position of the 
neutral axis. If a is the intensity of either the tensile or 
compressive stress at the distance unity from the neutral 
axis, then by the second of the preceding fundam.ental 
assumptions the intensity A", at any other distance z from 
the same axis or line of no stress, will be N = az. Again, 
if A is the area of the normal section of the beam, dA will 
be the area of an indefinitely small portion of that section, 
so that the amount of internal stress acting on it will be 
az.dA. If this differential amount of stress be integrated 
for the entire section, the preceding condition of equilibrium 
for either portion of the beam requires that the sum repre- 
sented by that total integration shall be equal to zero; 
or if d^ and d represent the distances of the most remote 
fibres on either side of the neutral axis, the following 
equations may be written: 



I'azdA =a I WA =o, 
J-d J-d 



or 

rd 

zdA =o . . (i) 



/:• 



• Eq. (i) shows that the static moment of the entire 
section about the neutral axis is equal to zero, and there- 
fore that the neutral axis passes through the centre of 
gravity or the centroid of the normal section. 

It is next necessary to determine the expression for 
the bending moment of the internal stresses of any sec- 
tion, such as JF of Fig. i , which is induced by and must 



Art. 14.] THE COMMON THEORY OF FLEXURE. 53 

be equal to the moment of the external forces acting upon 
either one of the two portions into which the beam is 
divided by that section. 

In Fig. I, let mn represent a differential length, dl 
of the neutral surface, and let p represent the radius of 
curvature of dl after flexure, also as shown in Fig. i, C 
being the centre of curvature. If u is the direct or longi- 
tudinal strain of a unit length of fibre at the distance unity 
from the neutral axis, when stressed with the intensity a, 
the strain in dl under that intensity will be tidl. BD is 
drawn parallel to JF, and represents the position of BD 
before flexure. The triangle D'mD ic, therefore, similar 
to Cmn. Consequently there may be written 

udl dl 1 

-T=-P' ■••"=? (^) 

Or the rate of strain, i.e., the strain of a unit length of 
fibre at distance unity from the neutral axis, is equal to 
the reciprocal of the radius of curvature. 

By the fundamental law or assumption of the common 
theory of flexure already given 

z 
Rate of strain at distance z=~. 

Then, by the fundamental law between stress and 
strain, the intensity A^ of the direct stress at any distance 
z is 

N=E-=Euz (3) 

If h is the variable breadth of section, the differential 
of the total stress is 

E 
Nbdz=~.(hdz).z (4) 



54 FLEXURE. [Ch. II. 

The differential moment of the internal stresses about 
the neutral axis will be 

dM=N.hdz.z=^~.{hdz).z^; .... (5) 

:.M=~ {hdz).z'=^- .... (6) 
pJ-d p 

in which / is the moment of inertia of the section of the 
beam about the neutral axis. 

If X is the horizontal coordinate of the neutral sur- 
face, and w the deflection or sag of the beam at any point, 
as indicated in Fig. i, when the curvature is small 



I d'^w 
P 



dx' 



and 



."-^'1? ■ (» 



Eq. (7) is the fimdamental equation by which the de- 
flection of a bent beam is found, whatever may be the 
character or amount of the loading. As indicated, it is 
strictly true only when the deflections are small ; in other 
words, when they are produced by strains within the elastic 
limit of such beams as are ordinarily used in engineering 
practice. That equation is easily integrated in all ordi- 
nary cases, if the value of the external bending moment M 
is expressed in terms of x, as will be abimdantly illustrated 
in succeeding articles. 

Another equation of great practical value remains to 
be established. Let it first be observed that the intensity 
of stress a, at the distance of unity from the neutral sur- 



Art. 14.J THE COMMON THEORY OF FLEXURE. ' 55 

face of a bent beam is a =Eu, by Hooke 's law, and further 
by eq. (2) 

a=Eu=~ (8) 

P 

' If the value of — from eq. (8) be vSubstituted in eq. (6) 

there will result 

M=al . (9) 

If the greatest intensity of stress in a normal section 

of a bent beam at the distance (i^ from the neutral axis be 

k 
represented by k, then ^ = v, and eq. (9) will take the form 

^'-T^ (-) 

Eq. (10) is one of the most important equations in the 
whole subject of the resistance of materials in consequence 
of its frequent use in the practical operation of designing 
beams or girders. Its employment is rendered exceed 
ingly simple and convenient by tables in which may be 
found computed the moments of inertia / for all the rolled 

sections, as well as values of the quantity -j, called the 

" section modulus." These tables are found in the various 
'* Hand-books " published by steel-producing companies, 
and they obviate essentially all num.erical computations 
for the determination of either moment of inertia or section 
modulus. Other tables may also be found which give the 
moments of inertia of a great variety of built sections, 
i.e., composite sections formed of various commercial 
rolled shapes such as plates, angles, channels, and I beams. 
In all the preceding expressions where the quantity 



56 FLEXURE. [Ch. II. 

Al appears it is to be taken to represent the bending mo- 
ment of the external forces, including the reactions, applied 
to a beam, the moment being taken about the neutral 
axis of the section under consideration. This external 
moment must necessarily be equal to the moment of the 
internal stresses represented by the last members of the 
preceding moment equations involving the greatest in- 
tensity of stress k of the section and the moment of inertia 
/ of the latter. 

There are one ' or two approximate features involved 
in the preceding analysis, the character of which is not 
discoverable when the fundamental laws of the theory of 
flexure are assumed rather than demonstrated, but which 
appear plainly evident in the true demonstration of the 
theory of flexure in App. I. It is obvious that the com- 
pression produced at the exterior surface of a bent beam 
at the points of loading is neglected or ignored in the pre- 
ceding demonstrations; but this does not sensibly affect 
the accuracy of the formulae which have been reached. 
There is, however, one result of the assumptions made 
which materially affects the accuracy of the formulas of the 
common theory of flexure for comparatively short beams. 
If the accurate analysis be followed it will be found that 
the formulae of that theory involve in reality the further 
assumption that the depth of the beam, i.e., in the direc- 
tion of the loading, is small in comparison with the length 
of span. The limit of ratio of length of span to depth 
above which the formulae may be applied with strict accu- 
racy cannot be definitely assigned, but there are many 
beams, especially of timber, emplo^^ed in engineering 
practice which are much too short in comparison with 
their depths to permit an accurate application of the for- 
mulae of the common theory of flexure. This observation 
bears with special' emphasis on computations for pins in 



Art. 15.] THE DISTRIBUTION OF SHEARING STRESS. 57 

pin -connected bridges which are treated as short beams. 
As a matter of fact, the common theory of flexure cannot 
be apphed to such short thick beams with any degree of 
accuracy whatever. It is, however, entirely permissible 
to use these fonnulae as general expressions, even under 
such loosely approximate conditions, into which empirical 
quantities established under the actual conditions of use 
are introduced, but they are not to be used in any other 
way. By such a procedure the formulas of the common 
theory of flexure have become of inestimable value to the 
civil engineer, but it is imperative to realize under what 
conditions , they may be employed with strict accuracy 
and under what conditions the introduction of quantities 
established by practical tests is required. 

Art. 15.^ — The Distribution of Shearing Stress in the Normal 
Section of a Bent Beam. 

The longitudinal fibres of a beam under loading take 
their stresses of tension and compression from the shearing 
stresses which are induced on vertical and horizontal 
planes in the interior of the beam. In order to realize 
what takes place in the interior of a beam let it be sup- 
posed to be divided into an indefinitely large number of 
small rectangular portions like those shown in the up- 
per part of Fig. i, and on a somewhat larger scale in the 
lower part. The vertical loading and reactions induce 
transverse shears, i.e., shearing stresses, on vertical trans- 
verse planes, which, as known from the general theory of 
stresses in solid bodies, induce shears of equal intensity on 
horizontal planes. The result is that which is shown in 
the lower portion of Fig. i. On the faces of the indefi- 
nitely small rectangular portions of the beam there are 
induced shears in pairs having the same intensity and act- 



58 



FLEXURE. 



[Ch. II. 



ing either toward or from a given edge. Each horizontal 
layer of the beam is, therefore, made to slide a little over 
the adjoining layers above and below it, as shown at A and 
A' in the lower part of Fig. i. 



Wi Wo w. 



w= w« 



OOP ,0,0 O 



i^//////M/.'A 



\: 



Wi Wo W3 W4 W5 

c , O O O -Q— Q 



Fig. I, 



Carefully remembering these general conditions, let the 
bending moment in the section ad of the beam in Fig. 2 
be represented by M and let the total transverse shear at 




Fig. 2. 



the same section be represented by 5. Then if x m.easured 

M 
horizontally from the section ad be so taken that ^="5. 



Art. 15.] THE DISTRIBUTION OF SHEARING STRESS. 59 

and if the intensity of the direct stress of tension or com- 
pression at the distance z from the neutral axis be repre- 
sented by ky there may at once be written 

M-=Sx=-\ :. k=-j-x (i) 

z 1 

k is thus seen to be a function of both z and x. If z be 
unchanged while x varies, the small variation of k for an 
indefinitely small variation of x will be 

Tx'^^'^T'^'' (2) 

If 5 is the intensity of the transverse shear at the dis- 
tance z from the neutral axis, the variation of that intensity 
for the indefinitely short distance dz {x remaining unchanged) 

ds 
will be -j~dz, and if the breadth or width of the beam is 6, 

the variation of longitudinal shear on the small horizontal 
area hdx for the small distance dz will be 

ds 

■^dzihdx) (3) 

The small shear given by expression (3) is equal to the 
variation of k f oimd by multiplying the members of eq. (2 ) 
by hdz, hence 

-j-dz.bdx=-jrdx.bdz\ (4) 

, ds Sz . ^ T , V 

••JJ^T' °^ ds = jzdz (5) 

It is obvious that the intensity of the shear, at the ex- 
terior surface of the beam is zero; in other words, s=o, 
when z=d the distance of the extreme fibre of the section 



6o FLEXURE. [Ch. II. 

from the neutral axis. Hence eq. (5) must be integrated 
between the Hmits of z and d, and that integration will 
give 

zdz~id'-z'). . . . (6)* 



u 



* The intensity of shear 5 is sometimes found with a partial regard only 
to the laws of the Common Theory of Flexure. In Fig. 3 the piece abed of 
a beam subjected to flexure whose neutral surface is NN is held in equilib- 
rium by the direct stresses on the faces be and ad in combination with the 
longitudinal shear on the face de. If ab is equal to dx and if y be the normal 
distance of any fibre from NN, obviously the difference between the direct 

stresses on the two sides be and ad will be ) dk.bdy in which b is the vari- 

Jy 

able width of the section. By the common theory of flexure, however, 

dM fyi 

tX ■ 

intensity of shear on the face dc the following equation at once results: 



dk = —j-y- Hence the above expression becomes ^^ | " ybdy. If s is the 



a b 



LV 



Fig. 3. 

, , dM Ai , , , , 

sodx = -^ I ybdy, (a) 



rijj''"''^'^ify- 



This equation differs from eq. (6) in that b, considered as a variable, 
appears in the second member. If the section is rectangular, b is constant 
and eq. (6) at once results. In fact if yi and y be taken as consecutive in 
eq. (a), which is the differential method of establishing s, that equation will be- 
come 

dsbdx = -Y-ybdy. 

The quantity b now disappears from the equation whether the width of the 
section be considered constant or variable. Then dividing both sides of the 



Art. 15.] THE DISTRIBUTION OF SHEARING STRESS. 61 ' 

The intensity s has its maximum value where z=o, i.e., 
at the neutral axis ; hence 

(max.)^=-^ (7) 

., , . . , , 8^^' 2bd' 

' If the section is rectangular / = = ■ 

12 3 

and 

(---•) ^4-i&- . • • . • (8) 

In other words, the maximum intensity of shear found at 

3 
the neutral axis is -, the average shear of the entire section. 

It is to be remembered that this intensity of shear s, 
Sit all points in the entire beam, acts on both the vertical 
and horizontal planes, i.e., this shear acts on longitudinal 
or horizontal planes parallel to the neutral surface as well 
as upon the vertical section of the beam. 

Eq. (6) is the equation of a parabola with its vertex 
in the neutral surface. Hence if Of be laid off, as shown 
in Fig. 2, at any convenient scale to represent the maxi- 
mum value of s, as given in eq. (7), and if from / as ver- 
tices the two branches of parabolic curves fa and fd be 
described as shown, any horizontal abscissa of the curves 
drawn from the line ad will represent the intensity of shear 
at that point. The origin of coordinates for eq, (6) is at 
in Fig. 2. 

equation by dx and integrating, eq. (6) of the text will be established. This 
means that all fibres equidistant from the neutral axis being stressed 
uniformly and hence without longitudinal shear along their vertical sides, 
the beam may be considered, so far as this analysis is concerned, as com- 
posed of vertical rectangular strips of width_ &, which may be of finite value 
or indefinitely small. 



62 



FLEXURE. 



[Ch. II. 



Distribution of Shear in Circular and Other Sections. 

A number of special approximate investigations have 
been made to determine the distribution of shear in the 
circular cross-section of a bent beam, involving more or 
less complicated consideration of stresses. While these 
investigations recognize the straight line variation of the 
intensities of normal stresses in the section under con- 
sideration, they are based on other conditions which are 




Fig. 4. 

not closely consistent with the fundamental assumptions 
of the Common Theory of Flexure. 

If the intensity of normal stress is the same at a uni- 
form distance from the neutral axis of the section, adjacent 
fibres equidistant from that axis will stretch the same 
amount, eliminating all shearing stresses between such 
fibres. If therefore a circular section whose area is A 
be divided into vertical strips each with the width dy 
as shown in Fig. 4, and if the notation shown in that 
figure be observed, eq. (6) may be adapted to the circular 
section by placing in the second member of that equation, 

5 — :. — for 5 and dy for I, resultine: as follows: 

A 12 '^ ^ 



=^v-d^) 



(9) 



Art. 15.] DISTRIBUTION OF SHEAR. 63 

This equation gives the value of the intensity of shear 
in all parts of the circular section. If z=d, i.e., at all points 
of the surface, the intensity 5 is zero. The maximum 

intensity is found by making 2 = 0, giving ^=—3, i.e., the 

maximum intensity of shear is | the mean, as was to be 
expected. The same result will necessarily follow the 
same mode of treatment of any form of section whatever^ 
as each such section is assumed to be made up of vertical 
rectangular strips between which no shear exists. The 
difference between this simple approximate method based 
upon results for a rectangular section and one of the 
special analyses for a circular section is shown by the 
maximum intensity of shearing stress at the neutral sur- 
face being found equal to | (instead of f ) of the mean by 
one of those special methods. If, however, the ordinary 
assumptions of the Common Theory of Flexure are to be 
made at all the advantage or increased accuracy of such 
special or more complicated analyses is not obvious. 

With such material as timber, in the case of beams, 
the longitudinal shear represented by 5 in either eq. (7) 
or eq. (8) may be the governing quantity in design. The 
capacity of timber to resist shear along its fibres is com- 
paratively so small that where the spans are relatively 
short failure will take place by shearing along the neutral 
surface before the extreme fibres yield either in tension 
or compression. In the design of timber beams, there- 
fore, and in other similar cases, it is necessary to test by 
computation the maximum value of .y as well as to deter- 
mine the greatest intensity of tensile or compressive stress 
in the extreme fibres, as wiU be completely shown in a 
later article. 



64 



FLEXURE. 



[Ch. 11. 



Art. 1 6. — External Bending Moments and Shears in General. 

Beams subjected to pure bending only will be treated 
here. 

A beam is said to be non-continuous if its extremities 
simply rest at each end of the span and suffer no constraint 
whatever. 

A beam, is said to be continuous if its length is equal 
to two or more spans, or if its ends, in case of one span (or 
more) suffer constraint. 

A cantilever is a beam which overhangs its span, one 
end of which is in no manner supported. Each of the 
overhanging portions of an open swing bridge is a canti- 
lever truss. 



We W5 v/, 

1 O O O 



mr^Tl 



'4,- 



Wo 



w, 



iJJIi: 



Fig. I. 

Fig. I represents a beam simply supported at each end, 
carrying the loads W ^, W^, W^, etc. Let bending moments 
be taken for any section, as JF, at the distance x^ from 
the right-hand abutment, at which location the reaction 
R^ acts. The load W^ is at the distance x^ from the sec- 
tion, W2 at the distance x^, and W^ at the distance x^ from 
the same section, the last distance not being shown in 
the figure. The bending moment desired will be the 
following : 

M=R'x'-W,x^-W,x^-W^,. . . . (i) 



Art. i6.] EXTERNAL BENDING MOMENTS AND SHEARS.. 65 

This equation is typical of all external bending moments 
for a beam simply supported at each end, whatever may 
be the system of loading or its amount, or whatever may 
be the location of the section. This equation is frequently 
written in the following form : 

M^IWx (2) 

The summation sign indicates that the sum is to be 
taken of the products form.ed by multiplying each external 
force, whether loading or reaction, by its lever-arm or 
normal distance from the section in question. It is a 
common and convenient mode of expressing the general 
value of the bending moment in any case whatever. 

In eq. (i) the differentials of x^, x^, x^, and x^ are all 
equal, so that if that equation be differentiated, the first 
derivative of M will have the following form : 

^=R'-W,-W,^W, = IW=S. . . (3) 

It will be at once evident that 5 in eq. (3) is the total 
transverse shear in the section for which the bending 
moment M is written, since the algebraic sum of R' and the 
loads between the end of the beam and the section con- 
stitutes that shear. Indeed, the usual manner of deter- 
mining the total transverse shear is the simple operation 
of summing up all the external forces acting on one of the 
portions of the beami formed by the section in question; 
the external forces, such as the reaction, acting in one 
direction being given one sign, and those, like the loading, 
acting in the other direction being given the opposite sign. 
The shear, therefore, becomes the numerical difference 
of the two sets of forces having opposite directions. 

Eq. (3) thus establishes the following important prin- 
ciple : The total transverse shear at any section is equal 



66 FLEXURE. [Ch.II. 

to the first differential coefficient of the bending moment con- 
sidered a function of x. 

In Fig. I the force 5 is supposed to be the resultant of 
the three loads W^, W^, and 14^3, and the reaction R\ i.e., 
the force 5 is supposed to represent that resultant both 
in line of action and magnitude. The bending moment M 
is, therefore, equal to Se, e being the normal distance of 
the line of action of 5 from the section, so that the actual 
bending moment upon any section of a bent beam may 
always be represented by the transverse shear, located 
as the resultant of all the external forces producing the 
bending moment, multiplied by its lever-arm. This is a 
simple but important matter of observation. 

In the section JF let the two equal and opposite 
forces 5 and — 5, numerically equal, act in opposite direc- 
tions; they will not, therefore, affect the equilibrium of 
the beam or any portion of it in any way whatever. As 
far as the equilibrium of the portion of the beam JF 
is concerned, the loads and the reactions may be supposed 
to be displaced by the couple 5, — S, with the lever-arm e, 
and the shear 5 acting upward in the section JF. The 
importance of this particular feature of the analysis con- 
sists in showing that in every bent beam carrying loads 
the action of the external forces (including the reaction) 
producing the bending is equivalent to a couple whose 
moment is Se acting about the neutral axis of the section 
I and the total transverse shear 5 acting in the section. 
The shear 5 evidently tends to move or slide one portion 
of the beam past the other, and an essential part of the 
operation of designing beams and trusses is its determina- 
tion at various sections with correspondingly various 
positions of loading. 

As is well known, the analytical condition for a maxi- 
mum or minimum bending moment in a beam is 



Art. i6.] EXTERNAL BENDING MOMENTS AND SHEARS. 67 

dM 



dx 



(4) 



From eqs. (3) and (4) is to be deduced the following 
principle: The greatest or least bending moment in any beam 
is to be found in that section for which the shear is zero. 

The greatest bending moment obviously is the only 
one of importance in the design of beams and trusses, and 
eq. (4) shows that the section in which it will be foimd 
can be located by simple inspection of the loading. It is 
onh^ necessary to sum up the reaction at one end and the 
loads adjacent to it, imtil the point is reached where the 
summation is zero. 'This point will usually be foimd 
where a load is supported. In that case the single load 
may arbitrarily be divided into two parts, supposed to act 
indefinitely near to each other, so that one of the parts 
may be just sufficient to make the zero summation desired. 
A single practical operation will make this feature per- 
fectly clear and simple. 

If the loading is uniformly continuous and of the 
intensity p, in each of the equations (i), (2), and (3) 
pdx is to be used for each of the separate loads W^, W^, W^, 
etc. The bending moment thus becomes 

M = Rx" - IWx = R'x' - r x.pdx^ R'x' - lpx\ (5) 

The expression for the shear then becomes 

^M^S^R'-P^. ..... (6) 

A second differentiation gives 

d'M 



dx' 



=-^. ...... (7) 



68 



FLEXURE. 



[Ch. II. 



Or, the second differential coefficient of the moment 
considered a function of x is equal to the intensity of the 
continuous load. 

This method of passing from, formula? for concentrated 
loads to those for continuous loads is perfectly simple and 
frequently employed. 



Art. 17. — Intermediate and End Shears. 

The beam shown in Fig. i is supposed to carry any 
loading whatever, and the figure is consequently intended 
to exhibit a uniform load in addition to a load of con- 
centrations. Inasmuch as all beams and other similar 
pieces have considerable weight, and sometimes great 
weight, ordinarily considered uniformly distributed over 
the span, this condition of loading is that which exists in 
all actual cases. The amount of uniform loading per 
linear unit, usually a foot, is represented by p, while the 



Wo 



* vit/^' 



W4 "5 Wg 



iz: 



-X 7-^B 

X- 



^^ 



1 

Fig. I. 

concentrations, as heretofore, are represented by VV^y W^, 
etc. 

The determination of the transverse shear at any sec- 
tion of a beam or truss is one of the m^ost frequent as well 
as one of the most important computations required in 
the design of structures. As has already been indicated, 
it is an extremely simple computation. It is first neces- 
sary, after knowing the position of the loading, to find the 
reactions at both ends of the span. In Fig. i the various 



Art. 17.] INTERMEDIATE AND END SHEARS. 69 

weights or loads are separated by the distances shown, a' 
being the distance from W^ to the reaction R or end of the 
span. IF(j is supposed to rest at the right end of the span 
for a purpose that will presently appear. The reaction 
7?" at the left end of the span (not shown) resulting from 
the concentrated loads only will have the following value : 



i?"=M 



\{^^^r-^) + iF.f- +^+,---+^ ) 



^^. ,'c + d-\-e\ j^,d-\-e ^^^e ^ ^ 
+ H'3( ^ l+W^, ^ +M^j. (i) 



The reaction B"' at t'le other end of the span (not 
shown) can be expressed ])y a similar equation, but it is 
simpler and more direct to write it as follows : 

R" ^ W, + W^ + W3 + W, + 1^5 - R"- . . (2) 

Obvious^ly the sum of the two reactions R" and R'" 
must be equal to the total concentrated loading. 

That part of the reaction due to the uniform load ex- 
tending over the span / will clearly be one half of that 
load or 

R^ = hpl-Rr ...... (3) 

The reaction R^ is supposed to be found at the left 
end of the span- and R^_ at the right end. The total re- 
actions then will be as follows. At left end of the span : 

R=R'' + ipl (4) 

At right end of the span : 

R'-^R'^' + ipl (5) 



/o FLEXURE. [Ch. II. 

The transverse shear at any interm.ediate section of 
the beam whatever may now readily be written. Let the 
section AB at the distance x from the left end of the span 
first be considered. The total loading between that sec- 
tion and the end of the span is W^ + W^ + px, and it acts 
downward. As the reaction R acts upward the expression 
for the shear will be 

S^R-W.-W.-px (6) 

In this case the section considered has been taken 
between two weights; let the section at the weight Wr, 
be considered, that weight being at the distance x^ from 
the end of the span. The amount of uniform load over 
the length x^ is simply px\ but inasmuch as the weight W^ 
is located at the section under consideration, the portion 
of that weight which may be taken as resting on the left 
of the section considered is indeterminate. In such cases 
it is proper and customary to take any portion or all of 
the weight as resting on either side of the section, but 
indefiaitely near to it. If it is a case where the maximum 
shear is desired, the single weight should be taken in such 
a position as to make the transverse shear as great as 
possible. If the case is one where it is desired to find the 
section at which the total load Irom that section to the 
end of the span is equal to the reaction, any portion may 
be taken which is found necessary to make the equality. 
If, for instance, px^ + W^ + ]\\ is less than R while px^ + 
W^ + W^ + W^ is greater than R, then that portion of U'3 
which would be considered on the left of the section but 
indefinitely near to it would be R — px^ — ]l\ — ]]\. The 
remaining portion of W^ would be considered as resting 
at the right of the section but indefinitely near to it. In 
such a case the transverse shear is zero at the weight l\\. 



Art. 17.] INTERMEDIATE AND END SHEARS. 7^ 

Again, let it be desired to find the greatest upward 
shear at W^, it being supposed that R is greater than the 
total load between W^ and the left end of the span. In 
this case no portion of W^ would be considered as acting 
to the left of the section, but the expression for the shear 
would be 

S^R-px'-W,-W, (7) 

It can be seen from the preceding statements that the 
maximum transverse shear in the beam will occur at the 
ends of the span where the value of the shear is the end 
reaction. Inasmuch as the end reaction R or R^ is thus 
the greatest shear in the entire span, it is a most important 
quantity to determine in the design of beams and trusses; 
it is the most important single factor in the determina- 
tion of the amount of material required at the end sections 
of both beams and trusses. The value of this end shear 
is given by the values for R and R' in eqs. (4) and (5). 

Since the total transverse shear in any section of a 
beam is simply the summation of all the external loads, 
including the reactions from one end of the span up to the 
section considered, it is evident, first, that that summation 
may be made from either end of the span, and second, 
that the amoimts so found will be equal numerically but 
affected by opposite signs. In determining the shear, 
therefore, in any given case, it is usual to make the sum- 
mation from that end of the span which can be used with 
the greatest convenience in computation. 

Fig. 2 exhibits a graphical representation of the pre- 
ceding treatment of intermediate and end shears, MN 
being the length of span shown in Fig. i. MF is the 
reaction R laid off at a convenient scale. The weights or 
loads Wj, W^, W^, etc., are laid off vertically downward in 
their proper locations at the same scale, as shown. The 



72 



FLEXURE. 



[Ch. II. 



vertical distance of G below F is the amount of tiniform 
load pa' between R and Wi in Fig. i, also laid down by the 
same scale. GGi is, therefore, the shear in the beam of 




Fig. 2. 



Fig. I immediately to the left of W^, and Hfi^ is the shear 
immediately to the right of the same load. Similarly, 
H^^H being drawn horizontally, HK is the amoimt of uniformi 
loading pa between W^ and W^. The remainder of the 
diagram is drawn in the same manner. 

Any vertical ordinate drawn from AIN either up or 
down to the broken line FGH^K ... represents the shear 
at the corresponding point in the span at the same scale 
used in laying off the reactions and loads. QQ^ is the shear 
at the point or section of beam at Q^, while TT. is the 
shear at the section T. The shear is zero at W^ where it 
changes its sign. At that point also will be found the 
greatest bending moment in the beam. 

As the diagram is drawn the shears on the left of W^ 
and above MN are positive, those on the right of W^ and 
below MN being negative ; but the diagram might have 



Art. 17.] 



INTERMEDIATE AND END SHEARS. 



73 



been drawn with equal propriety so as to have made R' 
and the shears between it and W^ positive and those be- 
tween that load and R negative. 

A glance at the diagram shows that the end shears, 
equal to the reactions, are the greatest in the span. 



h 










>/V 


1 










+ R 












W2 




M 










Ws N 


//m''^ 


1 


Ml 1 
























7/////// 


1 




w, 
























-R' 










(A/ 


5. 






Fig. 3. 



If a beam carries a load of concentrations only its shear 
diagram will be illustrated by Fig. 3, in which there are 
five loads, the diagram being composed of rectangles only. 
If, again, the load is wholly uniform Fig. 4 w^ill represent 
the shear diagram composed of two triangles with their 
apices at C, the centre of the span and point of no shear. 
Any vertical ordinate drawn from MN in either figure 



M 


MTh^c 




N 


W////, 


7 "^Li 




W 


W 


I ^<\\ 


HI 






Fig. 4. 

to the stepped line in the one case and to the straight line 
in the other will represent the shear at the section of beam 
from which the ordinate is drawn. Those diagrams repre- 



74 FLEXURE. [Ch. II. 

sent completely the graphical treatment of shears in all 
cases. 



Art. i8.— Maximum Reactions for Bridge Floor Beams. 

Three transverse floor beams of a railroad bridge are 
represented in Fig. i separated by the two spans /^ and / 
which, in a bridge, represent the panel lengths. The 
members .-I B and BC supporting the weights IF^, W^, 
etc., indicate the stringers which carry the railroad track 
and the train. The two beams or stringers AB and BC 
are considered simple non-continuous beams resting on 
the floor beams, but not necessarily nor usually on their 
tops. The problem is to determine the position of the 
locomotive or other train loads on the adjacent two short 
3pans l^ and /, so that the reaction R on the floor beam 
between shall have its greatest value. 

In Fig. I let a section of the beam be shown at R, and 
let X and xi be measured from the right ends of the two 
spans as shown in Fig. i, while Wi, W2, . . . 1^4 repre- 



Wo" 






R 

W3 W4 W5 

-^--e-^--c^— {+ 



-x---^ 



h ^1' ~ ~~ ~ I2 

Fig. I. 

sent a train of weights or wheel concentrations passing over 
the two spans from right to left. If R' and R are the 
reactions at A and B, respectively : 

h 



Art. i8.] MAXIMUM REACTIONS FOR BRIDGE FLOOR BEAMS. 75 

Then if the moments of weights and reactions be taken 
about C at the right-hand end of span h : 

R\h +« - {Wia+{Wi -\-W2)x) - {Wi +W2) (b -xi) 
-(VVi-{-W2-{-W3)c-{Wi+. . .+W4)d 

-I Wx-\-Rl2=o. (2) 

Hence, since R'h is equal to the quantity within the 
second parenthesis of the first member of eq. (2) : 

(PFia + (P^i+Pi^2)^)r-(^i-i-W^2)(6-:^i)-(P^i+VF2+1^3)c 
-{yVi+. . .^W^)d-Twx+Rl2=o (3) 

In order that the reaction R may have its greatest 
value it must remain unchanged when a small move- 
ment of the train is made. If therefore x-\-^x and xi-\-^x 
be written for x and xi, respectively, in eq. (3) and if eq. (3) 
be subtracted from the result so obtained, the following 
equations will be found : 

h 



h W^i+W2+etc. 



h+h '/v 



. (4) 



Eq. (4) shows the position of loading for the greatest 
value of the reaction R. It means simply that the ratio 
between the amount of loading on span h and the total 
load on both spans shall be the same as the ratio between 
the span h and the sum of the two spans (/1+/2). Inas- 
much as the load may move in either direction h may 



76 . . FLEXURE. [Ch. II. 

be written for h in the numerator of the first member 
of eq. (4). 

Clearly the two weights Wi and W2 in the preceding 
equations represent all the loads resting on span h whether 
there be two such weights or any number whatever. Sim- 
ilarly the weights indicated by the summation sign in the 
second member of eq. (4) represent the total load on both 
spans. If /i=/2, as is usually the case, the first m.ember 
of eq. (4) has the value of one-half. 

As in all such cases there may be more than one posi- 
tion of the loading which will satisfy the criterion eq. (4) ; 
in that case it is necessary to determine which- of those 
conditions will give the maximum of the ** greatest values " 
oiR. 

Inasmuch as the sum of the weights on the span h 
does not change for any value of xi equal to or less than 
b, it follows that a weight may be taken at the point of 
support B in satisfying eq. (4). This will simplify the use 
of eq. (3) in writing the expression for R. If xi=b there 
may at once be written from eq. (3) : 

-{Wia+{W, + W2)b)f-\-{Wi+W2 + Ws)c+{W,+ . . . +W,)d+ I Wx 
^ = ' k ^ ^5) 

This equation gives the value of R desired, and it is 
so written that numerical values may readily be computed 
by the use of tables. If h^h, as is usual, the ratio of 
those two quantities becomes unity. 

Art. 19. — Greatest Bending Moment Produced by Two , 
Equal Weights. 

Fig. I represents a non-continuous beam with the span / 
supporting two equal weights P, P. These two weights or 
loads are to be kept at a constant distance apart denoted 
by a. 



Art. 19.] BENDING MOMENT PRODUCED BY TWO WEIGHTS. 77 

It is required to find that position of the two loads 
which will cause the greatest bending moment to exist 
in the beam, and the value of that moment. The reac- 
tion R is to be found by the simple principle of the lever. 
Its value will therefore be 

l-(x + - 
R- \ ' .^P (i) 

Since the reaction. R can never be equal to 2P, IP, 
or the shear, must be equal to zero at the point of applica- 
tion of one of the loads P. In searching for the greatest 



j^ ^ — Tr)"'"ct) 



Fig. I. 

moment, then, it will only be necessary to find the moment 
about the point of application of one of the forces P. It 
will be most convenient to take that one nearest R. 
The moment desired will be 

M=Rx-=2P[x-j-—^] (2) 

dM ^/ 2X a 

= =2P I 



dx \ I 2lr 

1 a 
.*. x = . 

2 4 



This value in eq. (2) gives 



^^-^--i '> 



78 


FLEXURE, 


Since 






d'M 4P 

dx' ~ r 



[Ch. II. 



it appears that M^ is a maximum. 

The shear 5 in the section RP of the span will be the 
reaction R as given by eq. (i) : 

2P/ a 



5 = 2P-^(^^ + -) ; (4) 

Throughout the section a the shear 5' will be 

S'=5-P=P-f (.4-^). .■ . . . (5) 

Finally, between the right abutment and the nearest 
weight the shear 5^ will be 

5,=5_,p=_,?^(, + |). .-. . ..(6) 

If the separating distance, a, between the two weights 
be increased a value may be reached so great as to make 
the bending moment of the pair of weights less than that 
produced by placing one of them at the centre of the span. 
This limiting value of a may easily be found. The moment 
at the centre of span produced by placing a single weight 
P there is 

P I PI 



224 



By using eq. (3) 



PI P 
M'=M,; .•.-=-(/-^). ... (7) 



Art. 20.] BENDING MOMENTS OF CONCENTRATED LOADS. 79 

By solving this equation 

a=/(2-\/I)=.586Z (8) 

Whenever, therefore, the separating distance a is equal 
to or greater than .586 span length, the moment should 
be found by placing a single weight P at the centre of the 
span. 

Art. 20.— Position of Uniforn? Load for Greatest Shear and 
Greatest Bending Moment at any Section of a Non- 
Continuous Beam — Bending Moments of Concentrated 
Loads. 

A continuous load of uniform density is frequently 
employed in structural operations botli for beams and 
trusses, and it is essential to place such a load so as to 
produce the greatest effect both for shears and mom^ents. 
The position of loading for the greatest shear will first be 

found. 

A continuous train of any given uniform density ad- 
vances along a simple beam of span I. It is required to 
determine what position of loading will give the greatest shear 
at any specified section. 

In Fig. I, CD is the span /, and A is any section for 



m^, 



Fig. I. 



which it is required to find the position of the load for the 
greatest transverse shear. The load is supposed to ad- 
vance continuously from C to any point B. Let R be the 



8o FLEXURE. [Ch. II. 

reaction at D, and IP the load between A and B. The 
shear S' at A will be 

R-IP=^S' (i) 

Let R' be that part of R which is due to IP, and R" 
that part due to the load on CA. ; evidently R' is less than 
IP. Then 

R'-\-R''-IP=S\ 

If .45 carries no load, R' and i'P disappear in the value 
of 5. Hence 

is the shear for the head of the. train at A. 5 is 
greater than S' because IP is greater than R\ But no load 
can be taken from AC without decreasing R'\ Hence the 
greatest shear at any section will exist luhen the load extends 
from the end of the span to that section, whatever he the den- 
sity of the load. 

In general, the section will divide the span into tw^o un- 
equal segments. The load also m.ay approach from either 
direction. The greater or smaller segment, then, may be 
covered, and, according to the. principle just established, 
either one of these conditions will give a maximmm shear. 
A consideration of these conditions of loading in connec- 
tion with Fig. I, however, will show that these greatest 
shears will act in opposite directions. 

When the load covers the greater segm.ent the shear is 
called a main shear ; when it covers the smaller, it is called 
a counter shear. 

The determination of the greatest bending moment 
at any section .4 of a beam or truss, exemplified b}^ Fig. i, 
traversed by a continuous train of unifonri density is a 
very simple matter. It is clear that every part of the 



Art. 20.] BENDING MOMENTS OF CONCENTRATED LOADS. 8i 

uniform load resting on the beam will produce bending at 
any section considered ; and it is further obvious that every 
part of that uniform loading will create a bending moment 
at A of the same sign. It follows, therefore, that the 
entire span should be covered by the uniform train in order 
to produce a maximum bending, moment at any section 
of the beam or truss, and that this single position of the 
train will give the maximum bending moment throughout 
the entire span. 

The preceding position of moving load is taken only for 
a train of uniform density or for a series of uniform con- 
centrations, each pair of which is separated by the same 
distance as every other pair, i.e., for a uniformly distributed 
system of uniform concentrations. 

The general case of a simple beam loaded with any 
system of weights may be represented by Fig. 2, in w^hich 
the beam carries three loads ]V\, W\, and W^, spaced as 
show^n. The reactions or supporting forces R and R^ are 
detemiined in the usual manner by the law of the lever. 
Hence 

„ ^^.d d + c d + c + b ^ ^ 

R = W,j + W,-j- + W, J . . . . (2) 

A similar value may be written for R', but it is simpler 
after having found one reaction to w^rite 

• R'^W\ + W, + W,-R (3) 

The beam itself being supposed to have no weight, the 
bending moments at the points of application of the loads 
will be 

M^=Ra, 

AI,=R{a + b)-Wfi, \ . . (4) 

Ad,=R(a + b + c)- ]]\ {b + c) - IT' V- 



82 



FLEXURE. 



[Ch. II. 



After substituting the value of 7^ from eq. (2) in 
eqs. (4) the moments in the latter equations will be com- 
pletely known. 




S=-R 



Fig. 



The bending moment produced by each weight will be 
represented by the ordinates of the triangles shown in Fig. 2, 
the resultant moments at the points of application of the 
weights being given by eqs. (4). The ordinate CD repre- 
sents J/j in eqs. (4) by any convenient scale. Similarly 
FH represents M.^ in eqs. (4), and KL, M^. The hues 
AC, CF, FK, and KB are then drawn. Any vertical inter- 
cept between AB and the polygon ACFKB, found in the 
manner explained, will represent the bending mom.ent 
at the point where the intercept is drawn, and to the scale 
at which M^, M^, and M^ are laid down. This intercept is 
simply the sum of the intercepts of the triangles, each 
representing the partial bending moment due to a single 
weight. 



Art. 21.] BENDING MOMENT IN A NON-CONTINUOUS BEAM. 83 

Obviously the bending moments of any number of loads 
of any magnitude or of a uniform load, even, may be treated 
or represented in the same manner. 

The lower portion of Fig. 2 is the shear diagram drawn 
precisely as explained for Fig. 3 of Art. 17. 

Art. 21. — Greatest Bending Moment in a Non-Continuous 
Beam Produced by Concentrated Loads. 

The position of the moving load for the greatest bend- 
ing moment at any section of a non-continuous beam may 
be very simply determined. In Fig. i, let FG represent 
any such beam of the span /, and let any moving load what- 
ever, as W^ . . . Wn' . . . Wn advance from F toward G. 
Let G be the section at which it is desired to determine the 
maximum bending moment, and let n^ loads rest to the 
left of G, while n is the total number of loads on the span. 
Finally, let x' represent the distance of Wn' from G and to 
the left of that point, while x is the distance of Wn to the 
left of F. If a is the distance between W^ and W^_, b the 
distance between W., and IF3, c the distance between W^ 
and W4, etc., the reaction R at G will be 

^^.a + b + c+ ... -{-X 

w^ 1 

b + c + . . . +x 

^ . . . , (i) 



R=-i 



+ W, 



+11-;] 

The bending moment M about G will then take the 
value 



84 



FLEXURE. 



[Ch. II. 



M = Rr 



+ VK,( 6 + C+ . .. +^0 



Or, after inserting the value of R from above, 

M ^ ~[W,a + {W, + IF,)6 + (IF, + IF, + M/3)c 

+ . . . +(H/, + IF, + IF3+ . . . ^Wn)x\ \, ^ (2) 
- W,a - (IF, + IF,)^ - (IF, + W, + IF3)c: 

If the moving load advances by the amount Ax, the 
moment becomes, since Ax = Ax\ 



Kae-^ 



(^ O (yfi) 



0.0 



C.G. 

Fig. I. 



M' = Af + y (IF, + I/F2 + I/Fg + . . . + IF J i^ 



(IF,+IF,+ ... +H^n')^^. (3) 



Hence, for a maximum, the following value must never 
be negative : 

M'-.U==Ja: jy(IF, + lF2 + PF3+ ... +IFJ 

-(M/, + IF,+ ... +IF,0! =0. (4) 
Or the desired condition for a maximum takes the form 
U__ n\-MF3+_^.^+I;F,,.__ 
/"IF, + IF, + IF3+'... +IFn' • • • ^5) 



Art. 21.] BENDING MOMENT IN A NON-CONTINUOUS BEAM. 85 

It will seldom or never occur that this ratio will exactly 
exist if Wn' is supposed to be a whole weight; hence Wn' 
will usually be that part of a whole weight at C which is 
necessary to be taken in order that the equality (5) may 
hold. ■ 

, It is to be observed that if the moving load is very 
irregular,, so that there is a great and arbitrary diversity 
among the weights VV, there may be a number of positions 
of the moving load which will fulfil eq. (5), some one of 
which will give a value greater than any other; this is 
the absolute m.aximum desired. 

From what has preceded, it follows that Wn^ may 
always be taken at the point C in question; hence x^ in 
eq. (2) may always be taken equal to zero when that 
equation expresses the greatest value of the moment. The 
latter may then take either of the two following forms : 

M =j[W,a + (W, + W,)b + . . . + (W, + W, ^ 

+ . . . + Wr,)x] - W,a - {W, + W,) b 
- ... -(I^, + T^3+ ... +iy .._,)(?) 

M=j[W,(a + b+ . . . +:r)+F/,(5 + c;+ . . . +x) 

+ W,(c + d+ . . . +x)+ . . . +WnX] 
^W,(a + b+ . . . +?)-W,{b+ ...+?) 
- ... -Wn'..(?) 



(6) 



(6a) 



In these equations x corresponds to the position of 
maximum bending, while the sign (?) represents the dis- 
tance between the concentrations Wn'-i and Wn'. 

The preceding equations give the greatest bending 
moments at any arbitrarily assigned points in the span. 
There remains to be determined the point at which the 
•greatesi! moment in 'the entire span exists, and the mag- 
nitude of that greatest moment. 



86 FLEXURE. " [Ch. II. 

It has already been shown that for any given condition 
of loading the greatest bending moment in the beam will 
occur at that section for which the shear is zero. But if 
the shear is zero, the reaction R must be equal to the sum 
of the weights (I/F1+I/F2+. . .-\-Wn') between G and C, 
the latter now being the section at which the greatest 
moment in the span exists. 

Hence for that section eq. (5) will take the form 



/' R 



I Wi+W2-^Wz-{-. . .+Wn 



(7) 



Hence 



V 



R^j{Wi+W2+. . .+Wn). ... (8) 



The relations existing in eqs. (7) and (8) can obtain 
only if the centre of gravity CG in Fig. i is at the dis- 
tance V from F, showing that the centre of gravity of the 
load is at the same distance from one end of the beam 
as the section or point of greatest bending is from the 
other. In other words, the distance between the point of 
greatest bending for any given system of loading and the 
centre of gravity of the latter is bisected by the centre of span. 

If the load is uniform, therefore, it must cover the whole 
span. 

It is to be observed that eq. (6) is composed of the sums 
H^,, 1^1 + 14^2' ^"tc, multiplied by the distances a, 6, c, etc. 
Again, as in the equation immediately preceding eq. (2), 
the expression for the moment, M, may be taken as com- 
posed of the positive products of each of the single weights 
Wi, W2, etc., multiplied by its distance from any point 
distant x to the right of W„ and of the negative products 
similarly taken in reference to the section located by x', 
as shown byeq. (6a). 



Art. 21.] 

IMIII 



" 11 1 1 1 1 1 1 1 1 11 1 i 1 1 1 1 1 1 1 1 1 1 n 1 1 r ! 1 1 1 1 1 1 1 1 



240 



7o.O 13 

630^ 



lOJ.O 18 

iiro . 



13i.O 23 

1860 



lui.6 32 

2485 . 



174.0 37 

3205. 



193.0 431213.0 4Si22 

4040 



\m 



iM^ 



4 5 



6 7 8 9 

^ o'(R) 6' (^ 5-(JM) 



24550 



101 

22910 



17000 



14^0 



77 

11G90 



10190 



8790 



18 



13 



10 



22420 



411.0 9G 381.0 91 

17980 



351.0 86 

15250 



321.0 81 

12670 



291.0 

■ 10240 8830 



£52.0 61 iJ 

7530 



391.5 91 301.5 80 

18900 16170 



331.5 81 

13590 



11160 



271.5 67 252.0 

8880 7570 



232.0 50 

6360 



172.0 85 

16670 



342.0 80 312.0 75 

14120 11720 



232.0 70 

9470 



252.0 61 232.5 50 

7370 6180 



iiao 30 
5090 ' 



16^0 



14910 



J22.5 75 292.5 70 

12500 102.50 



262.5 65 

8150 



232.5 50 

6200 



213.0 51 

5110 



4120 



13090 



333.0 '71 

11900 



303.0 00. 

9780 



273.0 01 

7800 



243.0 56 

5970 



2ia0 47 

4290 



193.5 42 

3370 



174.0 30 i: 

2.350 ' 



318.0 74 

11500 



303.0 00 

10400 



73.0 01 

8410 



24a0 50 

6580 



U3.0 51 

4900 



183.0 42 103.5 37 

3370 25.55 



144.0 31 1- 

ia30 ■ 



10060 



273.0 01 

9030 



243.0 50 

7200 



213.0 51 183.0 46 

5520 3990 



153.0 37 

2005 



33.5 32 

1885 



114.0 26 

1283 



8770 



243.0 50 
7810 



iliO 51 

6130 



183.0 40 153.0 

4600 



123,0 32 

1992 



103.5 

1368 



228.0 50 213.0 43 183.0 43 153.0 33 123.0 

6950 6110 4670 3380 2240 



33 93.0 24 73.5 19 54.0 13^34. 

1248 780 409.5 I 



Loads and moments are for one raU 

Loads given in thousands of pounds 

Moments « » " " foot pounds 

Moments are expressed to a limit of error of 0.1 per cent 




LEI. 



I I I II II I I I I I 11 I I I I I I I I I I M I I I II I I I I riTl I I I II I ! 11 I I I I I I I I I 




203.0 64 23S.0 69U13.0 74 34S.0 79 

7740 ^1 9810^1 12030^ l^UOO^ 



367.5 88 

16100 



7.0 93 

17930. 



19850 



99 426.0 104 

21900. 



10-1 

1© 



11-2 12-3 13-4 



14-5 



30)) -'((30)) -'((30)) -/(l30} 



15-6 16-7 



17-8 18-9 



300 



340 



>70 



110 



>40 



550 



>37 



32 



6310 



213.0 43 

5240 



193.5 43 

4280 



174.0 37 

3230 



154.5 32 

2460 



1245 



5514 



(3.0 40 

4524 



159.0 29 

2678 



139.5 24 

1980 



120.0 ] 

900 



105.0 13 

720 



5.0 13 

345 



120 



90.0 10 

450 



G0.0 

150 



4164 



103.0 35 

3325 



2965 



38.0 3( 

2275 



143.5 3J 113.5 2 
1682 



129.0 2 
1808 



109.5 19 

1260 



90.0 10 

450 



oao 
150 



690 



00.0 

150 



1914 



108.0 25 

1374 



982 



518 



270 



1014 



rao 10 
624 



53.5 11 

331.5 



)7.5 



605 



58.5 11 39.0 

312 97.5 



.10 5 

292.5 97.5 



1.0 6 

117 



MOMENT TABLE 

COOPER'S E-60 LOADING 

Two 213-ton Engines + 6000 lbs. p.l.ft. 

Scale: 1"=15' 



{To face page 87.) 



Art. 21.] BENDING MOMENT IN A NON-CONTINUOUS BEAM. 87 

The practical application of the preceding formulae 
can therefore best be effected by means of a tabulation of 
moments like that shown in Table I, taken from the stand- 
ard specifications of the N. Y. C. R. R. Co. for 191 5. The 
wheel weights and train loads shown in the table are for 
one rail only, i.e., they are half those for one track. By 
comparing the weights and spacings with those in Fig. i 
and eq. (6) it will be seen that 1^1 = 15,000 lbs.; W2 = 
30,000 lbs.; W3= 2,0,000 lbs., etc., and that a = 8 ft.; b = 
5 ft. ; <: = 5 ft., etc. 

The arrangement of Table I essentially as shown has 
been used for a long time to expedite the computations of 
moments and shears produced by wheel concentrations, 
followed by a heavy uniform load. It will be noticed that 
the first line at the top of the diagram shows the progress- 
ive sums of the individual loads beginning at the left- 
hand end, i.e., at Wi, in connection with the progressive 
sums of the distances between the centres of each pair 
of wheels. The second line (in the larger figures) is the 
progressive sums of the moments of the wheel loads about 
the centre of Wi, i.e., i860 is the moment of W2, W3, 
Wa, and W5 about the centre of Wi. Each of the hori- 
zontal spaces below the heavy line on which the wheel 
concentrations rest contains one line of small figures and 
one line of large figures. The small figures are the pro- 
gressive sums of the distances from the head of the uniform 
moving load or from each successive wheel to each of the 
wheel weights in the series. The larger figures give the 
progressive sums of the moments of the wheel weights 
beginning with l^is about the head of the uniform load, 
i.e., 19.5 X5 =97-5. and 19.5 XioH-97.5 =292.5. Each hori- 
zontal space is seen to begin at the vertical heavy line 
under each weight taken in succession and to contain the 
progressive sums of the moments, weights, and distances 



88 FLEXURE. 



[Ch. 11. 



about or from each such weight, as is clear on examining 
the diagram. At the left of each horizontal line there is 
found the number of the wheel load under which the right- 
hand end of the line begins. 

The diagrammatic exhibit of these various numerical 
quantities w^ll enable the reactions, shears, and greatest 
moments at any point in the span to be readily deter- 
mined. 

When a uniform train load is a part of the system 
of loading it is only necessary to consider any section 
of it as acting through its centre of gravity, i.e., through 
its mid-point. Taking that centre as its point of appli- 
cation the separating space is the distance from that point 
to the nearest concentration. If in Table II 20 ft. of 
train load be used, that train weight will be 60,000 lbs. 
applied at the distance 10 + 5=15 ft. from load 18. This 
simple operation is all that is needed for any uniform 
load or for a series of sections of uniform load. 

Table II is a table of maximum moments, end shears* 
and floor-beam reactions for girders having spans up to 125 
ft., and it is taken from the New York Central Railroad 
Specifications for 191 5. The shears and floor-beam reactions, 
like the results shown in Table I, are given in thousands 
of pounds and are for one rail only. The moments are given 
in thousands of foot-pounds, like the moments shown in 
Table I. The loading is the same as that shown by the 
diagram in Table I, except that the results for spans up 
to a maximum of 11 ft. are found by using a special 
loading of two 72,000-lb. axle loads 7 ft. apart, or 36,000 
lbs. for each rail. The maximum moments are found for 
the conditions of loading given by the criterion, eq. (5), 
of this article. The maximum floor-beam reactions are 
found by eq. (5) of Art. 18, in accordance with the 
criterion, eq. (4), of the same article. 



Art. 21.] BENDING MOMENT IN A NON-CONTINUOUS BEAM. 89 

Table II. 
TABLE OF MAXIMUM MOMENTS, END SHEARS AND FLOOR- 
BEAM REACTIONS FOR GIRDERS. 

Moments in Thousands of Foot-pounds. 

Shears and Floor-beam Reactions in Thousands of Pounds. 

Loading Two E 60 Engines and Train Load of 6000 lbs. per Foot or Special 
Loading Two 72,000-lb. Axle Loads 7 Ft. C to C. 

Results for One Rail. Results from Special Loading Marked *. 



Span. 
Ft. 



Maximum 
Moments. 



10 
II 
12 
13 
14 

15 
16 

17 

18 

19 

20 
21 
22 

23 
24 

25 
26 
27 
28 
29 

30 
31 
32 

33 
34 



* 450 

* 54-0 

* 63.0 

* 72.0 

* 81.0 



■90.0 
= 99.0 
120.0 

142.5 
165.0 

187.5 
210.0 

232.5 
255.0 
280.0 

309 -5 
339.0 
368.5 
398.2 
427.8 

457-5 
487.2 
516.9 

548.3 
582.0 

615.8 

649 -3 
683 . 2 
716.9 
750.6 



End 
Shear. 


Floor- 
beam 
Reaction. 


Span. 


*36.o 


*36.o 


35 


*36.o 


40.0 


36 


38.6 


47.1 


37 


41-3 


52.5 


38 


*44 


56.7 


39 


*46.8 


60.0 


40 


49.1 


65.5 


41 


52.5 


70.0 


42 


55-4 


73-9 


43 


57.8 


78.2 


44 


60.0 


82.0 


45 


63.8 


85.3 


46 


67.1 


88.2 


47 


70.0 


91 .0 


48 


72.6 


94-3 


49 


75 


98.3 


50 


77.1 


101.9 


51 


79.1 


105.2 


52 


80.9 


108.2 


53 


83.1 


no. 9 


54 


■85.2 


II3-5 


55 


87.1 


116. 6 


56 


.88.9 


120. 1 


57 


90.6 


123.4 


58 


92.3 


126.5 


59 


94.6 


129.4 


60 


96.6 


132.7 


61 


98.6 


136.5 


62 


100.4 


140.0 


63 


102. 1 


143-2 


64 



Maximum 
Moments. 



784.5 
823.0 
861.6 
900.0 
940.0 

983.4 
1027.0 
1070.4 
1113.9 

II57-4 

1201 . I 
1244.4 
1287.9 
1331-4 

1378.3 

1426.3 

1474-7 
1522.8 
1571.0 
1622.2 

1675.2 
1728.6 
1781.9 

1835- I 

1891 .4 

1949.4 

2007 . 5 
2065.4 
2123.4 
2183.3 



End 
Shear. 



103.8 

105.9 
107.8 
109.7 
III .4 

113. 1 

115. 2 
117. 2 

119. 

120.8 

122.5 
124.2 

125-9 
127-5 
129.2 

130.8 
132.5 
134- 1 
135-7 
137-4 

139.0 
140.6 
142.2 
143-8 
145-4 

147.0 
148.6 
150.2 
152.0 
153.8 



Floor- 
beam 
Reaction. 



146.4 

149-3 
152.2 
155.6 

158.8 

162.0 



90 



FLEXURE. 

Table U.—{Con.) 



[Ch. II. 



Span. 
Ft. 



Maximum 
Moments. 



End 
Shear. 



65 
66 

67 
68 
69 

70 

71 

72 

73 
74 

75 
76 
77 
78 
79 

80 
81 
82 
83 
84 

85 
86 

87 
88 

89 

90 

91 
92 

93 
94 



2246.3 
2309.3 
2372.3 

2435 -4 
2498.4 

2560.4 
2624.5 
2688.3 
2750.9 
2819.4 

2888.6 
2958.0 
3028.6 
3096.6 
3168.2 

3240.7 
33II-4 
3385-1 
3459-6 
3534-6 

3610.4 
3689.4 
3766.5 
3846.0 
3924-3 

4005 . 8 
4084 . 4 
4164.0 
4246.6 
4328.0 



155-7 
157-5 
159.6 
161. 7 
163.8 

165.8 
167.7 
170.0 
172.2 
174-4 

176.5 
178.6 
180.6 

182.5 
184.4 

186.3 

188. 

190. 

192. 

194. 

196. 
198. 
200. 
202. 
204. 



205.8 
207.7 
209.7 
211 .6 
213-5 



Floor 

beam 

Reaction. 



Span. 



95 
96 
97 
98 

99 
100 

lOI 

102 
103 
104 

105 
106 
107 
108 
109 

no 
III 

112 

113 
114 

115 
116 
117 
118 
119 

120 
121 
122 
123 
124 

125 



Maximum 
Moments. 



4408 . 4 
4490.7 

4573-5 
4659.8 

4743-8 

4830.0 
4916.9 
5004 . o 

5115-5 
5212.8 

5306.5 
5401.3 
5499 • 2 
5617.0 
5727.6 

5829.6 

5937-4 
6040 . o 
6148.2 
6258.0 

6366 . 8 
6478.0 
6586.1 
6696 . 6 
6808 . 3 

692 1 . 6 
7030.5 
7143-8 
7260.1 
7376.4 

7495 - 2 



End 
Shear. 



215-4 
217.2 
219.2 
221 .2 
223.1 

225.0 
226.8 

223.6 

230.4 
232.3 

234-1 
235-9 
237-7 
239-4 
241.2 

243.0 
244.8 

246.6 

248-3 
250.0 

251-8 
253-6 
255-3 
257.0 
258.8 

260.5 

262.2 
264.0 

265.7 
267.4 

269.1 



Floor. 

beam 

Reaction. 



Problem. 

Let a single-track railroad plate girder v/ith an effective 
span of 88 ft. be traversed from right to left by the 
moving load shown in Table I. It is required to find 
the greatest bending moments and shears at the centre 



Art. 21.] BENDING MOMENT IN A NON-CONTINUOUS BEAM. 91 

and quarter-points of the span, the dead load or own 
weight of the girder, floor system and track being taken 
at 1800 lbs. per linear foot. 

Dead Load. 

By eq. (6) of Art. 22 the bending moments at the 
quarter-point and centre are, since the reaction R is 
44X900 =39,600 lbs.; 

Quarter-point. Centre. 

X=jl = 22 it. X=^l = 44 ft. 

M ^-{Ix — x^) 654,000 ft. -lbs. 871,000 ft. -lbs. 

2 

By eq. (7) of Art. 22, the shears at end, quarter- 
point, and centre are: 

End. Quarter-point. Centre. 

X=0 X = 22ft. x=/[/\.ii. 

Shear = 39,600 lbs. 19,800 lbs. zero 

Moving Load. 

If weight W4. be placed at the quarter-point of the 
span, 14 wheel weights will rest on the girder with Wi4. 

I' 
4 ft. from the right-hand end of the span. As j=\, 

1/ 

,. .^ . / \ • -^1 l^ 75,000 105,000 

the criterion, eq. (s), gives either -r=-^ or, -—^ , 

I 367,500 367,500' 

the first being too small and the second too large. Hence 
W4: at the quarter-point is the proper position for the 
maximum bending moment. Wi will be 84 ft. from the 
right-hand end of the span. Taking moments of all the 
wheels about that point, by the aid of Table I, the reac- 
tion R at the left end of the span is : 

„ 14,830,000 .^ ., 
R= ^' = 168,500 lbs. 



92 FLEXURE. [Ch. II. 

Eq. (6) will then give the bending moment at VF4, 
but having the reaction R and using Table I the bending 
moment becomes: 

M = 168,500 X22 —720,000 = 2,987,000 ft. -lbs. 

The end shear with the load placed so as to produce 
the greatest bending moment at the quarter-point is ob- 
viously the reaction i? = i68,5oolbs. The shear immedi ately 
at the left of the quarter-point will be 1-68,500 — 75,000 
= 93,500 lbs. 

The greatest bending moment at the centre of span 
is similarly found. If Wo be placed at the centre of the 
span the wheel weights Wi . . . Wn will rest on the 
span, the latter being 3 ft. to the left of the right-hand 
end of the span. The ratio representing the criterion, 

eq. (s) is - = ^^ or — -. The first of these values is too 
/ 258 258 

large and the latter is too small, showing that W5 at the 

centre of the span is the correct position for the greatest 

bending moment at that point. The reaction R for this 

position of the load is at once written by the aid of Table 

I as follow. 

R = ^7y° + l5^X2 X 1000 = 108,000 lbs. 

The bending moment M for the centre of the span is 
as follow^s, using the preceding value of R and Table I : 

M = (108 X44 — 1245) Xiooo =3,059,000 ft. -lbs. 

The end shear for this position of the loading is the 
reaction R, i.e., 108,000 lbs. The shear indefinitely near 
to but at the left of the center is 108,000 — 105,000=3000 
lbs. This small shear shows that the moment at the cen- 



Art. 21.] BENDING MOMENT OF A NON-CONTINUOUS BEAM. 93 

tre of the span is the greatest in the entire span for this 
position of loading. 

Assembling the preceding results, the total dead and 
moving load moments and shears will be as follows : 

Moments. 

Quarter-point. Centre. 

Dead Load 654,000 ft. -lbs. 871,000 ft. -lbs. 

Moving Load. . . .2,987,000 ft. -lbs. 3,509,000 ft. -lbs. 

3,641,000 ft. -lbs. 4,380,000 ft. -lbs. 

Shears. 

End. Quarter-point. Centre. 

Dead Load 39,600 lbs. 19,800 lbs. zero 

Moving Load. . . . 168,500 lbs. 93,500 lbs. 3000 lbs. 

Total.. 208,100 lbs. 113,300 lbs. 3000 lbs. 

The expression " equivalent uniform load," for moments 
or shears, as the case may be, is sometimes used. It 
simply means that the uniform load is such as to produce 
the moments or shears equivalent to those found under 
given conditions. A uniform load p per linear foot acting 

on the entire span / will produce a centre-moment of ^. 

8 

. pP 

Hence if there be written ^ =3,905,000, then, if / =88: 



p= X3, 509, 000 =3625 lbs. per linear foot. 

7744 

The equivalent uniform load therefore for the greatest 
bending moment at the centre of the span is 3625 lbs. 
per linear foot. Similarly as the bending moment at any 



94 FLEXURE. [Ch. 11. 

distance x from one end of the span is -{lx—x^)/iix be made 

2 

2 2 in the present case, / being ^d> feet, there will be found 
by placing this expression equal to 2,987,000 ft. -lbs: 



p= \ =4.114 lbs. per linear foot. 

726 



The end shear for a uniform load over the whole span 
is equal to the load on half the span. Hence by placing 
^ X44 = 108,000 lbs., there will result: 



't) = '- = 2455 lbs. per linear foot. 

44 



This is the equivalent uniform load for the end shear 
with the load so placed as to give the greatest bending 
moment at the centre of the span. 

In the same way the equivalent uniform load for the 
end shear 168,600 lbs., with the load placed so as to give 
the greatest bending moment at the quarter-point, will 
be found to be 3830 lbs. per linear foot. 

These simple instances show that the equivalent uni- 
form load varies from one case to another according to 
the amount, distribution and position of the loading. 

Art. 22. — Moments and vShears in Special Cases. 

Certain special cases of beams are of such common 
occurrence, and consequently of such importance, that a 
somewhat more detailed treatment than that alread) 
given may be deemed desirable. The following cases are 
of this character:- 




Art. 22.] MOMENTS AND SHEARS IN SPECIAL CASES. 95 

Case I. 

Let a non -continuous beam supporting a single weight 

P at any point be con- 
sidered, and let such a 
beam be represented in 
Fig. I. If the span RR' 
is represented by 

Fig I. l=a-\-h=RP-\-RT, 

the reactions i^ and R' will be 

R=^P, and R'^-^P (i) 

Consequently, if x represents the distance of any sec- 
tion in RP from R, while x' represents the distance of any 
section of R'P from R\ the general values of the bending 
moments for the two segments a and 6 of the beam will be 

M = Rx, and M^ =R'x' (2) 

These two moments become equal to each other and 
represent the greatest bending moment in the beam when 

x-=a and x' = b, 

or when the section is taken at the point of application of the 
load P. 

Eq. (2) shows that the moments vary directly as the 
distances from the ends of the beam. Hence HAP (nor- 
mal to RR^) is taken by any convenient scale to represent 

the greatest moment, ~r^, and if RAR^ is drawn, any 

intercept parallel to AP and lying between RAR and RR' 
will represent the bending moment for the section at its 
foot by the same scale. In this mariner CD is the bend- 
ing moment at D. 

The shear is imiform for each single segment; it is 



96 



FLEXURE. 



[Ch. II. 



evidently equal to R for RP and R^ for R^P. It becomes 
zero at P, where is found the greatest bending moment. 

Case II. 

Again, let Fig. 2 represent the same beam shown in 
Fig. I, but let the load be one of uniform intensity, /?, 
extending from end to end of the beam. Let C be placed 
at the centre of the span, 
and let R and R\ as before, 
represent the two reactions. 
Since the load is symmetri- 
cal in reference to C, 

R=R. 
For the same reason the 
moments and shears in one Fig. 2. 

half of the beam will be exactly like those in the other; 
consequently reference will be made to one half of the 
beam only. Let x and x^ then be measured fromi R 
toward C. The forces acting upon the beam are R and 
p, the latter being uniformly continuous. Applying the 
formula for the bending m.om.ent at any section x, re- 
membering that x^ has all values less than x, 




M=Rx-p / 



x;)dx^\ 



M = Rx- 



px' 



(3) 



If / is the span, at C, M becomes 

Rl pP 

But because the load is uniform 

2 



(4) 



Art. 22.] MOMENTS AND SHEARS IN SPECIAL CASES, 97 

Hence 



if W is put for the total load. Placing 
in eq. (3), 






M=Hlx-x') (6) 

The moments M, therefore, are proportional to the 
abscissae of a parabola whose vertex is over C, and which 
passes through the origin of coordinates R. Let AC, then, 
normal to RR\ be taken equal to M^, and let the parabola 
RAR' be draAvn. Intercepts, as FH, parallel to AC, will 
represent bending moments in the sections, as H, at their 
feet. 

The shear at any section is 

s='^4^=R-p^=pi--^)' .... (7) 



dx 



or it is equal to the load covering that portion of the beam 
between the section in question and the centre. 

Eq. (7) shows that the shear at the centre is zero; it 
also shows that 5=i? at the ends of the beam. It further 
demonstrates that the shear varies directly as the distance 
from the centre. Hence, take RB to represent R and draw 
BC. The shear at any vsection, as H, will then be repre- 
sented by the vertical intercept, as EG, included between 
BC and RC. 

The shear being zero at the centre, the greatest bending 
moment will also be found at that point. This is also 
evident from inspection of the loading. 

Eq. (2) of Case I shows that if a beam of span / carries a 



98 



FLEXURE. 



[Ch. 11. 



W 
weight — at its centre, the moment M at the same point 

will be 

W I Wl 



M,= 



(8) 



The third member of eq. (8) is identical with the third 
member of eq. (5). It is shown, therefore, that a load 
concentrated at the centre of a non-continuous beam will 
cause the same moment, at that centre, as double the same 
load tmiformly distributed over the span. 

Eqs. (5) and (8) are much used in connection with the 
bending of ordinary non-continuous beams, whether solid 
or flanged; and such beams are frequently foimd. 

Case III. 

The third case to be taken is a cantilever imiformly 
loaded; it is shown in Fig. 3. Let 
X be measured from the free end A, 
and let the uniform intensity of the 
load be represented by p. The load 
px acts with its centre at the distance 
^x from the section x. Hence the 
desired moment will be 

px^ 



M=-px--- 

2 2 



^- . (9) 




Fig. 3. 



If AB = /, the moment at B is 

^ 2 



(10) 



The negative sign is used to indicate that the lower side 
of the beam is subjected to compression. In the two pre- 
ceding cases, evidently the upper side is in compression. 

The shear at any section is 






(11) 



Art. 23.) FORMULA OF COMMON THEORY OF FLEXURE. 99 

Hence the shear at any section is the load between the free 
end and that section. 

Eq. (9) shows that the moments vary as the square 
of the distance from the free end; consequently the 
moment curve is a parabola with the vertex at .4, and 
with a vertical axis. Let BC, then, represent il/j by any 
convenient scale and draw the parabola CD A. Any ver- 
tical intercept, as DF, will represent the moment at the 
section, as F, at its foot. 

Again, let BG represent the shear pi at B, then draw 
the straight line AG. Any vertical intercept, as HF, will 
then represent the shear at the corresponding section F. 



Art. 23. — Recapitulation of the General Formulae of the 
Common Theory of Flexure. 

It is convenient for many purposes to arrange the 
formulse of the Common Theory of Flexure in the most 
general and concise form. In this article the preceding 
general formulse for shear, strains, resisting moments, and 
deflections will be recapitulated and so arranged. In 
order to complete the generalization, the summation sign 2 
will be used instead of the sign of integration. 

In Fig. 1, let ^^C represent the centre line of any bent 
beam; y4/% a vertical line through A; C/%a horizontal line 
through C, while A is the section of the beam at which the 
deflection (vertical or horizontal) in reference to C, the 
bending moment, the shearing stress, etc., are to be deter- 
mined. As shown in figure, let x be the horizontal coor- 
dinate measured from A, and y the vertical one measured 
from the same point ; then let Xi be the horizontal distance 
from the same point to the point of application of any 
external vertical force P. To complete the notation, let D 



lOO 



FLEXURE. 



[Ch. II. 



be the deflection desired; Mi, the moment of the external 
forces about A\ S, the shear at A\ u, the strain (exten- 




FlG. I. 

sion or compression) per imit of length of a fibre parallel to 
the neutral surface and situated at a normal distance of 
imity from it ; /, the general expression of the moment of 
inertia of a normal cross-section of the beam, taken in 
reference to the neutral axis of that section ; E, the coeffi- 
cient of elasticity for the material of the beam ; and M the 
moment of the external forces for any section, as B. 

Again, let J be an indefinitely small portion of any 
normal cross-section of the beam, and let z be an ordinate 
normal to the neutral axis of the same section. By the 
" common theory " of flexure, the intensity of stress at the 
distance z from the neutral surface is (zP'E). Conse- 
quently the stress developed in the portion i of the sec- 
tion is EP'zA, and the resisting moment of that stress 
is EP'z'^-A. 

The resisting moment of the whole section will there- 
fore be found by taking the sum of all such moments for 
its whole area. 

Hence 

M = EuIz''^=EuI. 

Hence, also, 

M 
''=EI' 



Art. 23.] FORMUL/E OF COMMON THEORY OF FLEXURE. 10 1 

If n represents an indefinitely short portion of the 
neutral surface, the strain for such a length of fibre at unit's 
distance from that surface will be nu. 

If the beam were originally straight and horizontal, n 
would be equal to dx. 

u being supposed small, the effect of the strain mi at 
any section, B, is to cause the end A of the chord BA to 
move vertically through the distance nux. 

If BK and BA (taken equal) are the positions of the 
chords before and after flexure, mix will be the vertical 
distance between K and A. 

By precisely the same kinematical principle the ex- 
pression nuy will be the horizontal movement of A in 
reference to B. 

Let Inux and Imty represent summations extending 
from A to C, then will those expressions be the vertical and 
horizontal deflections respectively of A in reference to C. 
It is evident that these operations are perfectly general, 
and that x and y may be taken iii any direction whatever. 

The following general but strictly approximate equa- 
tions relating to the subject of flexure may now be written : 

S=IP (i) 

Mi=IPxi (2) 



''=eT ..-.••• (3) 



Imi=In—-:. o (4) 

EI ^ 



Dr, = Inux=I^^^ (5) 



102 FLEXURE. [Ch. II. 
D, = Snuy=l'^ (6) 

Dh represents horizontal deflection. 

The summation IPz must extend from A to a point of 
no bending, or from A to a point at which the bending 
moment is ill/. In the latter case 

M, = IPz-\-M,' (7) 

Ml may be positive or negative. 

Art. 24. — The Theorem of Three Moments. 

The object of this theorem is the determination of the 
relation existing between the bending mioments which are 
fotmd in any continuous beam at any three adjacent points 
of support. In the most general case to which the theorem 
applies, the section of the beam is supposed to be variable, 
the points of support are not supposed to be in the same 
level, and at any point, or all points, of support there may 
be constraint applied to the beam external to the load 
w^hich it is to carry ; or, what is equivalent to the last con- 
dition, the beam may not be straight at any point of sup- 
port before flexure takes place. 

Before establishing the theorem itself, some prelimi- 
nary matters must receive attention. 

If a beam is simply supported at each end, the reactions 
are foimd by dividing the applied loads according to the 
simple principle of the lever. If, however, either or both 
ends are not simply supported, the reaction in general is 
greater at one end and less at the other than would be 
found by the law of the lever ; a portion of the reaction at 
one end is, as it were, transferred to the other. The trans- 



Art. 24. 



THE THEOREM OF THREE MOMENTS. 



103 



ference can only be accomplished by the application of a 
couple to the beam, the forces of the couple being applied 
at the two adjacent points of support; the span, conse- 
quently, will be the lever-arm of the couple. The existence 
of equilibrium requires the application to the beam of an 
equal and opposite couple. It is only necessary, however, 
to consider, in connection with the span AB, the one shown 
in Fig. I. Further, from what has immediately preceded, 




Fig. 



it appears that the force of this couple is equal to the 
difference between the actual reaction at either point of 
support and that foimd by the law of the lever. The 
bending caused by this couple may evidently be of an 
opposite kind to that existing in a beam simply supported 
at each end. 

These results are represented graphically in Fig. i. A 
and B are points o^ support, and ^^ is the beam ; AR and 
BR' are the reactions according to the law of the lever; 
RF = R'F is the force of the applied couple ; consequently 

AF=AR + RF and BF =BR' - {R'F =RF) 

are the reactions after the couple is applied. As is well 
known, lines parallel to CK, drawn in the triangle ACB, 



104 



FLEXURE. 



[Ch. II. 



represent the bending moments at the various sections of 
the beam, when the reactions are AR and BR\ Finally, 
vertical lines parallel to AG, in the triangle QHG, will 
represent the bending moments caused by the force R'F. 

In the general case there may also be applied to the 
beam two equal and opposite couples having axes passing 
through A and B respectively. The effect of such couples 
will be nothing so far as the reactions are concerned, but 
the}^ will cause uniform bending between A and B. This 




Fig. 2. 







^ 




V 


D 




* N 


C 




/ 


V 


/ 


"^ 


f 


H 






Q 



Fig. 3. 

miiform or constant moment may be represented by ver- 
tical lines drawn jjarallel to AH or LA" (equal to each 
other) between the lines AB and HQ. The resultant 
moments to which the various sections of the beam are 
subjected Avill then be represented by the algebraic sum 
of the three vertical ordinates included between the lines 
ACB and GQ. Let that resultant be called M. This 
composition of the resultant moment M will be made 
clearer by reference to Figs. 2 and 3. Fig. 2 shows the 
component moment. due to the single force F acting with 



Art. 24.J THE THEOREM OF THREE MOMENTS. 105 

the lever-ami / so that its moment increases directly as 
the distance from B. Fig. 3, on the other hand, shows the 
component moment due to the two equal and opposite 
couples acting at the ends of the span. The resultant 
mom.ent M is the algebraic sum of the three component 
moments, shown combined in Fig. i. 

Let the moment GA be called Ma, and the moment 

BQ=^LN^HA=Ah. 

Also designate the moment caused by the load P, shown 
by lines parallel to CK in ACB, by M^. Then let x be any 
horizontal distance measured from A toward B; I the 
horizontal distance AB ; and z the distance of the point of 
application, K, of the force P from A. With this nota- 
tion there can be at once written 

A{=AIaC-^)+MJj)+AI, (i)* 



/ / ' ^'^'\l 



Eq. (i) is simply the general form of eq. (2), Art. 23. 

It is to be noticed that Fig. i does not show all the 
moments Ma, Mb, and M^ to be the same sign, but for 
convenience they are so written in eq. (i). 

The formula which represents the theorem of three 
moments can now be written without difficulty. The 
method to be followed involves the improvements added 
by Prof. H. T. Eddy, and is the same as that given by him 
in the "American Journal of Mathematics," Vol. I., No. i. 

Fig. 4 shows a portion of a continuous beam, including 
two spans and three points of supports. The deflections 
will be supposed measured from the horizontal line NQ. 
The spans are represented by la and k; the vertical dis- 

* This equation is used in the next Art. for a short demonstration of 
the common form of the Theorem of Three Moments. 



o6 



FLEXURE. 



[Ch. 11. 



tances of NQ from the points of support by c„, cj,, and c,; 
the moments at the same points by Ma, Mt, and M,, while 
the letters 5 and R represent shears and reactions re- 
spectively. 

In order to make the case general, it will be supposed 
that the beam is curved in a vertical plane, and has an 



N 

"I 

k- — ^ ^ 



'Ma 



^U 




Fig. 4. 

elbow at 6, before flexure, and that, at that point of sup- 
port, the tangent of its inclination to a horizontal line, 
toward the span la, is t, while f represents the tangent on 
the other side of the same point of support ; also let d and 
<i' be the vertical distances, before bending takes place, of 
the points a and c, respectively, below the tangents at the 
point b. 

A portion of the difference between Ca and Cb is due to 
the original inclination, whose tangent is t, and the original 
lack of straightness, and is not caused by the bending; 
that portion which is due to the bending, however, is, 
remembering eq. (5), Art. 23, 

D=Ca^Cb — lat — a = J> ~^pT ' 

Fig. 5 will make clear the corriponent parts of the value 
of D in the preceding equation. 

By the aid of eq. (i) this equation may be written: 

E{Ca — Cb — lat-d) 



Art. 24.] 



THE THEOREM OF THREE MOMENTS. 



107 



= /[{„,(t.>M.@.,,|f]. <., 

In this equation, it is to be remembered, both x and z 
(involved in M^) are measured from support a toward 




Fig, 5. 

support b. Now let a similar equation be written for the 
span /^, in which the variables x and z will be measured 
from c toward b. There will then result 
E{c^-c,-l/-d') 

=<[{«.(¥)-«'(f)+".ff]. « 



When the general sign of summation is displaced by 
the integral sign, n becomes the differential of the axis of 
the beam, or ds. But ds may be represented by udx, u 
being such a function of x as becomes unity if the axis of 
the beam is originally straight and parallel to the axis of x. 
The eqs. (2) and (3) may then be reduced to simpler forms 
by the following methods:* 



* These analytic transformations are of the nature of convenient but 
arbitrary notation and are not to any degree whatever analytic demon- 
strations. 



io8 FLEXURE. [Ch. II. 

In eq. (2) put 
'Ul — x\xn I f'^ u(l^ — x)xdx Xa f'^ u(la — x)dx 

Also 

Xa f'' u{la-x)dx ia^a f" ,j , , ' , 

Also 

irJb '^^(^a-x)dx=—j^ / {Ia-X)dx= — . (6) 

In the same manner 



^ x^n I f"^ ux^dx Xa /*^ uxdx 
Also 

Xa' fUixdx ia'xj C^ 

And 

"^a Xa I -J la Xa It. a I i ^n Xq ^o, 'a , \ 

— -, — / uxax= -J / xax= — . . (o) 

la J"^ la Jh 2 ^^^ 

Again, in the same manner, 

^Isl xn 
2 — 7— ^^i^aii^a^M^xAx (10) 

b 1 

Using eqs. (4) to (10), eq. (2) may be written: 

E{Cu-C^-lat-d) =- (MaUaiaXa + Miflif^idXa') 

+ li,ai^o ^l M^XJX. (11) 



Art. 24.] THE THEOREM OF THREE MOMENTS. 109 

Proceeding in precisely the same manner with the span 
Ky ^^- (3) becomes 

c 

-{-UiJ^^I M^xJx. (12) 

b 

The quantities Xa and x^ are to be determined by apply- 
ing eq. (4) to the span indicated by the subscript ; while 
^a, 4, ^^, and i^ are to be detennined by using eqs. (5) and 
(6) in the same way. vSimJlar observations apply to uj, 
4', Xa, ^^/,%\ and x/ taken in connection with eqs. (7), 
(8), and (9). 

If / is not a continuous fiinction of x, the various inte- 
grations of eqs. (4), (5), (7), and (8) must give place to 
summation: (I) taken between the proper limits. 

Dividing eqs. (11) and (12J by la and l^ respectively, 
and adding the results, 



^/Ca-Cb C^-Cb ^ d d\ 



(13) 



in which T = t + f. 

Eq. (13) is the most general form of the theorem of 
three moments if E, the coefficient of elasticity, is a con- 
stant quantity. Indeed, that equation expresses, as it 
stands, the ''theorem" for a variabL coefficient of elas- 
ticity if (ie) be written instead of i; e representing a quan- 
tity determined in a manner exactly similar to that used 
in connection with the quantity i. 



I lo FLEXURE. [Ch. 11. 

In the ordinary case of an engineer's experience T =o, 
d=d' =o, I = constant, u=u^^u^=etc., =c^ = secant of the 
inclination for which t = ~f is the tangent; consequently 

*^o "^a ^c c ^4a ^4C J' 



From eq. (4) 



From eq. (7) 



2^a ^h 






The stimmation IM^xAx can be readily made by refer- 
ring to Fig. I. 

The moment represented by CK in that figure is 

l-z 



consequently the moment at any point between A and K^ 
due to Py is 

«,-p('-=-'),.f=p(ti).. 

Between K and B 

Using these quantities for the span /^, 

a f*z ria 

IM^xJx= / M,xdx+ / M,'xdx = lP{l^^-z^)z. 



Art. 24.] THE THEOREM OF THREE MOMENTS. in 

For the span l^ the subscript a is to be changed to c. 
Introducing all these quantities eq. (13) becomes, aftei 
providing for any number of weights, P: 

■\-\^P{K'-z')z + }lP{i;-z')z. (14) 

Eq. (14), with d equal to imity, is the form in which the 
theorem of three moments is usually given; with c^ equal 
to unity or not, it applies only to a beam which is straight 
before flexure, since 

T = t-\-t'=o=a=d', 

If such a beam rests on the supports a, 6, and c, before 
bending takes place, 



a "c 

and the first member of eq. (14) becomes zero. 

If, in the general case to which eq. (13) applies, the 
deflections c^, c^, and c^ belong to the beam in a position 
of no bending, the first member of that equation disappears, 
since it is the sum of the deflections due to bending only 
for the spans l^ and /^, divided by those spans, and each 
of those quantities is zero by the equation immediately 
preceding, eq. (2). Also, if the beam or truss belonging 
to each span is straight between the points of support 
{such points being supposed in the same level or not) , u^ = 
uj =u^,^= constant, and u^=u/ =u^^= another constant. If, 
finally, / be again taken as constant, x^ and x^, as well as 
IM^xJx, will have the values found above. 

From these considerations it at once follows that the 



1 12 FLEXURE. [Ch. II. 

second member of eq. (14), put equal to zero, expresses 
the theorem of three moments for a beam or truss straight 
between points of support, when those points are not in 
the same level, but when they belong to a configuration 
of no bending in the beam. Such an equation, however, 
does not belong to a beam not straight between points of 
support. 

The shear at either end of any span, as /^, is next 
to be found, and it can be at once written by referring to 
the observations made in connection with Fig. i. It was 
there seen that the. reaction found by the simple law of 
the lever is to be increased or decreased for the continuous 
beam, by an amount found by dividing the difference of 
the moments at the extremities of any span by the span 
itself. Referring, therefore, to Fig. 4, for the shears 5, 
there may at once be written: 






S. = ^P^j v^^ (15) 



a 



z , M-M 



Sj = jp +-^ K ..... (16) 



/. / 



a 



f„2 M-M 



S, = IPj + ^ ' (17) 

X-z M^-M, 



s:=ip^^—^. — \ .... (18) 



The negative sign is put before the fraction 



in eq. (15) because in Fig. i the moments M^ and Mj^ are 
represented opposite in sign to that caused by P, while in 



Art. 24-] THE THEOREM OF THREE MOMENTS. 113 

eq. (i) the three moments are given the same sign, as has 
alreaci}^ been noticed. 

Eqs, (15) to (18) are so written as to make an upward 
reaction positive, and they may, perhaps, be more simply 
found by taking moments about either end of a span. For 
' example, taking moments about the right end of /^, 



SJ^-IP{l^-z)+M^=M,. 



From this, eq. (15) at once results. Again, moments 
about the left end of the same span give 

This equation gives eq. (16), and the same process will 
give the others. 

If the loading over the different spans is of uniform 
intensity, then, in general, P =wdz, w being the intensity. 
Consequently 

IP{P-z'')z= f w{P-z')zdz=w—. 
Jo 4 

In all equations, therefore, for 

chere is to be placed the term w^-^ ; and for 

-^IP{i;-z'')z 

I ^ 
the term w-^. The letters a and c mean, of course, that 

'4 

reference is made to the spans l^ and /^. 



114 FLEXURE. [Ch. II. 

From Fig. 4, there may at once be written: 

R =Sa'+Sa. ...... (19) 

R' =S,'+S, (20) 

R''=S/+Sc, ....... (21) 

etc.=etc. 4-etc. 

Art. 25. — Short Demonstration of the Conmion Form of the 
Theorem of Three Moments. 

The general demonstration of the Theorem of Three 
Moments given in the preceding article has the great 
advantage of showing the influence of all the elements 
which enter the complete problem, including variability of 
moment of inertia, lack of straightness of beam, and points 
of support not at the same elevation. An adequate con- 
ception of the influences of the assumptions made in estab- 
lishing the common or approximate form of the theorem 
can be obtained only by the employment of the general 
analysis, but it is convenient to establish the usual or 
approximate form of the theorem by a short direct method 
like the following. 

Eq. (i) of the preceding article gives the general value 
of the bending moment in any span whatever of a con- 
tinuous beam such as that shown in Fig. i. The notation 
given in that figure explains itself and is essentially the same 
as that already used. It should be remembered that each 
reaction R, R\ and R" is composed of two shears as indi- 
cated, one acting at an indefinitely short distance to the 
left of a point of support and the other at an indefinitely 
short distance to the right of the same support. It is 
supposed that one load acts in each span at the distance 



Art. 25.] COMMON FORM OF THEOREM OF THREE MOMENTS. 115 

z from the left-hand end of the left-hand span, or from 
the right-hand end of the right-hand span. 

Using eq. (i) of the preceding article and representing 
the deflection at any point in the span h by w, eq. (i) may 
be at once written : 

^^S^^«C-ir^)+^'^+^- ■ • • (X) 

The quantity /i is the moment of inertia of the cross- 
section of the beam about its neutral axis and E is the 



I 



A I M„ 



I 



B M 



C I Mc 



Fig. I. 



modulus of elasticity. It is assumed that the beam is 
straight and horizontal and that the moment of inertia 
does not vary in either span. If ti is the tangent of the 
inclination of the neutral surface of the beam at the right- 
hand end of the span h, then integrating eq. (i) between 
the limits of x and h eq. (2) will at once result: 



dx Ell \ h \ 2 2/2/1 



The integration of Midx is indicated only in eq. (2) 
for the reason that in general Mi is a discontinuous func- 
tion. The double integral j ' ( Midx^ cannot therefore 
generally be completed by the usual procedures, but it 



ii6 FLEXURE. [Ch. II. 

must be taken as —r^InMicc, as given by eq. (5) of Art. 

23. The value of this expression for a single load Pi is 
shown in detail on the lower half of page 1 10 of the preceding 
Art. as \Pi{li~ —z~)z, which appears in eq. (3). By integra- 
ting eq. (2) between the limits of h and 0, remembering 
that the points of support are supposed to be at the same 
elevation and hence that w = ior x = li\ 

w^ -^(Mali+2M,h+^(li'-z')z)+6h=^o. (3) 

An equation identical with eq. (3) may be written for 
the right-hand span h by simply changing the subscripts, 
remembering, however, that the origin from which z and x 
are measured is the point of support C, Fig. i , and that the 
tangent of the inclination of the neutral surface at the 
left-hand end of the span h will be —^1. 

Hence : 

w= -^(Mcl2-\-2M,l2+^(l2'-z'-)z) = -6h=o. (4) 

If eqs. (3) and (4) be added the usual and approximate 
form of the Theorem of Three Moments will at once result, 
except that the moments of inertia Ii and 1 2 are different. 
Assuming Ii^Io and writing the summation sign before 
Pi and P2 to indicate that any number of loads may act 
on every span, the Theorem of Three Moments as usually 
employed will at once result : 

MJi + 2M,(/i -I-/2) +Mc/2 = - r^Piili^ -z^)z 

n 

-~IP2(P2-Z')Z . . . . (S) 

h 



Art. 25.] COMMON FORM OF THEOREM OF THREE MOMENTS. 117 

It will be observed that eq. (5) is identical with the 
second member of eq. (14) of the preceding article, and it 
is the equation sought. The expressions for the shears com- 
posing each of the reactions may now easily be written. 

Taking moments about the right-hand end of the span /i : 

• Sah-^Pl(ll-z)-\-Ma=M, (6) 

Hence : 

^ ^p h-z Ma-Mb , , 

^'=^^'^, h~- ^^) 

Again taking moments about the left-hand end of the 
same span: 

S',h-IP,z+M,=Ma (8) 

Hence : 

^ 6 = ^/^1^ + - -. (9) 

Eqs. (7) and (9) give the shears at the two ends of the 
span l\ and they also give the shears at the two ends of 
the span I2 by simply changing the notation so as to apply 
to the span I2 as shown in eqs. (10) and (11) : 

bi = 2F2-r^ -. (10) 

Sc = ^P2—, , .... (11) 

/2 12 

Each reaction will be the sum of the appropriate pair 
of shears as shown by eqs. (19), (20), and (21) of the pre- 
ceding article. 

These equations are given in their most general forms; 



ii8 FLEXURE. . [Ch. 11. 

that is, for any disposition of loads of any magnitude. They 
may be adapted to uniform loading either partial or entire, 
as indicated on the lower half of page 113. 



Art. 26. — Reaction under Continuous Beam of any Number 

of Spans. 

The general value of the reactions at the points of 
support under any continuous beam have been given in 
eqs. (19), (20), (21), etc., of article 24. Before those 
equations, however, can be applied to any particular case, 
the values of the bending moments, which appear in the 
expressions Sa, Sb, 5^, etc., for the shears, must be deter- 
mined. In the application of the theorem of three mo- 
ments, it is usually assumed that the continuous beam 
before flexure is straight between the points of support, 
and that the latter belong to a configuration of no bending. 
The moment of inertia I is also assumed to be constant. 
This is frequently not strictly true, yet it will be assumed 
in what follows, since the method to be used in finding 
the moments is independent of the assumption, and remains 
precisely the same whatever form for the theorem of three 
moments may be chosen. 

Agreeably to the assumption made, eq. (5)* of the pre- 
ceding article takes the following form : 

Mala + 2M,{la+lc) +Mclc = -y^ PilJ" - Z^)z 



}sP{U^~z')z (i) 

Lc 



* Or eq; (14) of Art. 24. 



Art. 26.] REACTIONS UNDER ANY CONTINUOUS BEAM. 119 

Let Fig. I represent a continuous beam of n spans 
equal or unequal in length. At the points of support, 



Fig. I. 

o, I, 2, 3, 4, 5, etc., let the bending moments be represented 
by Mq, Mp ilf 2, M3, etc. The moment M^ is always known ; 
it is ordinarily zero, and that will be considered its value. 

An examination of Fig. i shows that, by repeated 
applications of eq. (i), the number of resulting equations 
of condition will be one less than the number of spans. 
If the two end moments are known (here assumed to be 
zero), the number of unknown moments will also be one 
less than the number of spans. Hence the number of 
equations will always be sufficient for the determination 
of the unknown moments. 

For the sake of brevity let the following notation be 
adopted : 



^s--r^P(h'-z')z-YlP(l,^-z')z. 
etc. = etc. — etc. 

b,=l,; c, = 2(l, + l,)', d,=l,. 
c,=h\ d, = 2(l, + l,); f,=l,. 



I20 FLEXURE. ]Ch. II. 

i den'^ting any number of the series i, 2, 3, 4, , . . n. It is 
thus seen that, in general, 

qi = 2{pi + Si)\ 

also that a^=h^, c^=h^, d^=c^, etc. These relations can be 
used to simplify the final result. 

By repeated applications of eq. (i) the following n 
equations of condition, involving the notation given above, 
will result: 

alM^+hlM2 =ui ^ 

a2Ml+b2M2-\-C2M3 =U2 

-\-b3M2-\-C3M3-i-d3M4: =U3 

+ C4iV/3+G^4M4+/4M5 =^4^* ^^^ 

-\-d5M4. +/5M5 +goM6 = U5 



= Un 



These simultaneous equations may be treated in various 
ways in order to determine the values of the moments Mi, 
M2, M3, etc. The preceding notation 'is adapted to the 
method by determinants, which is probably as simple as 
any. As these procedures are purely algebraic they will 
not be further developed here. 

In American engineering practice, as exemplified in the 
theory of revolving-swing bridges, it is necessary to con- 
sider at most, two simultaneous equations of condition 
whose solution requires the simplest process of elimination 
only. 



Art. 27.] DEFLECTION BY THE COMMON THEORY. 121 

This last case may be simply illustrated by referring 
to Fig. I, in which Mo =0. If there are three spans M3 =0 
as one of the end spans. The first two of eq. (2) will be 
needed : 

aiMi+6iM2=wi, ..... (3) 

a2Mi+&2M2=W2 (4) 

Simple elimination will then give: 

,^ b2Mi—biU2 J ,^ aiU2—a2Ui , . 

M\= — r r; and M2 = — r r-. . (5) 

ai02—a20i aib2—a20i 

Reactions. 

After the moments are found, either by the general or 
special method, for any condition of loading, the reactions 
will at once result from the substitution of the values thus 
found in the eqs. (15) to (21) of iVrt. 24, which it is not neces- 
sary to reproduce here. 

Art. 27. — Deflection by the Common Theory of Flexure. 

The deflection or sag of a beam subjected to loading at 
right angles to its axis is the displacement of the neutral 
surface in the direction of the loading. Ordinarily the 
beam is horizontal and the loading vertical, so that the 
deflection is also vertical. The entire deflection is due both 
to the lengthening and the shortening of the fibres on the 
two sides of the netural surface and to the action of the 
transverse shear throughout the beam. The equation 
leading directly to the former portion is eq. (7) of Art 14, 
but the equations of Art. 24 must be used to determine the 
deflection due to shear. 

Let xo be the coordinate of some point at which the 



122 FLEXURE. tch. ii. 

tangent of the inclination of the neutral surface to the axis 
of X is known ; then from eq. (7) of Art. 14 



dw 
dx 






dw 

-J- will be at once recognized as the general value of the 

tangent of the inclination just mentioned, or, in the case 
of curved beams, as approximately the difference between 
the tangent, before and after flexure. 

Again, let x^ represent the coordinate of a point at which 
the deflection w is known, then from eq. (i) : 



w= -^jdx' (2) 



Er 



The points of greatest or least deflection and greatest 
or least inclination of neutral surface are easily found by 
the aid of eqs. (i) and (2). 

The point of greatest or least deflection is evidently 
foimd by putting 

dw 

5^=^ (3) 

dw 
and solving for x. Since -j- is the value of the tangent of 

the inclination of the neutral surface, it follows that a 
point of greatest or least deflection is found where the beam 
is horizontal. 

Again, the point at which the inclination will be great- 
est or least is found by the equation 

\dx J (Pw 



Art. 27.] DEFLECTION BY THE COMMON THEORY, 123 

d^iv 
But, approximately, -t-t is the reciprocal of the radius 

of curvature; hence the greatest inclination will be found 
at that ■ point at which the radius of curvature becomes infi- 
nitely great, or, at that point at which the curvature changes 
from positive to negative or vice versa. These points are 
called points of "contra -flexure." Since: 

d'^w 

there is no bending at a point of contra-flexure. 

The moment of the external forces, M, will always be 
expressed in terms of x. After the insertion of such values, 
eqs. (i) and (2) may at once be integrated and (3) and (4) 
solved. 

The coefhcient of elasticity, B, is always considered a 
constant quantity ; hence it may always be taken outside the 
integral signs. In all ordinary cases, also, / is constant 
throughout the entire beam. In such cases, then, there 
will only need to be integrated the expressions: 



/x rx rx 

Mdx and / / Mdx' 
^ Xi J Xa 



It is sometimes convenient to express the tangent of 
inclination of the neutral surface and the deflection in 
terms of some known intensity k^ of fibre stress at the 
distance d from the neutral surface and at a section of the 
beam where the known external bending moment is M^. 
The desired expressions may readily be written by simply 
transforming eqs. (i) and (2) to the proper shape. It 

has been shown by eq. (10) of Art. 14 that k^= — j- , and 



124 FLEXURE, [Ch. II. 

hence that I = —r-. By substitution of this value of / 



k 



first. in eq. (i) and then in eq. (2), there will result: 



0^*^ Xq 



and 



w 



= c^ f' r^dx' (6) 

EMJJxi Jxo 



Eqs. (5) and (6) give the desired expressions in which 
/ and d are considered constant in accordance with all 
ordinary practice. In the use of these last two equations 
it is supposed that the conditions of any given problems 
will enable k^ and M^ to be computed as known quantities. 

The general form of the integral in the second member 
of eq. (6) is easily determined. The quantities M^^ and 
M are exactly similar expressions with the same number 
of terms and of the same degree. The effect of the inte- 
gration of M twice between the limits indicated is to raise 
the degree of each term of which it is composed by two, 
so that the double integration of Mdx^ divided by M^ will 
be a simple product aP, a being a numerical quantity 
depending upon the manner of loading, the condition of 
the ends of the beam, or other attendant circumstances of 
the same general character. Inserting these results in 
eq. (6), the expression for the deflection will become 

^^ ""Ed ^ ^^ 

Eq. (6a) is not often used, but there are some practical 
applications of formulae in which it must be employed. 



Art. 27.] DEFLECTION DUE TO SHEARING. 12 5 

Deflection Due to Shearing. 

That portion of the deflection due to transverse shear- 
ing may be determined as readily as that due to the length- 
ening and shortening of the fibres of the bent beam. In 
determining the requisite equations it is necessary to con- 
sider only the intensity of shear in the neutral surface, 
as it is the deflection of that surface which is sought. 

Let w' be the deflection due to shearing and let ^ repre- 
sent the transverse shearing strain for a unit of length of 
the beam. The transverse strain for an indefinitely short 
portion dx of the neutral surface will then be dw' = ^dx. 
If G represents the coefficient of elasticity for shear, while 
5 represents the intensity of shear, eq. (3) of Art. 2 shows 

that 9^ =7^. There may then be written: 

dw^ = (j)dx = y^dx (7) 

By using the value of 5 given in eq. (7) of Art. 15? 

dw' =—f-^dx (8) 

The general expressions for the shearing deflection 
will, therefore, take the form: 



w 






The integration required in eq. (9) can be made with 
ease in any given case, as it is necessary only to express 
the value of the total transverse shear 5 in terms of x. 
The application of that equation to special cases will be 



126 



FLEXURE. 



[Ch. 11. 



made in a later article. Obviously the total deflection in 
any bent beam will be the sum : 

w-\-w'. . (lo) 



Art. 28. —The Neutral Curve for Special Cases. 

The curved intersection of the neutral surface with a 
vertical plane passing through the axis of a loaded, and 
originally straight, beam may be called the "neutral 
curve." The neutral curve is the locus of the extremities 
of the ordinates w of Art. 27; it therefore gives the deflec- 
tion at any point of the beam due to the direct stresses of 
tension and compression in it, but not due to the effect of 
transverse shear, which will be treated in a subsequent 
article. 

The method of finding the neutral curve for any par- 
ticular case of beam or loading can be well illustrated by 
the operations in the following three cases: 

Case I. 

This case is shown in the accompanying figure, which 
represents a cantilever carrying a uniform load with a 



i 



< -X- H 



I 



Fig. I 



single weight W at. its free end. As usual, the intensitv 
of the uniform loading will be represented by p. 



Art. 28.] THE NEUTRAL CURVE FOR SPECIAL CASES. 127 

Measuring x and w from B, as shown, the general value 
of the bending moment is 



M-E,'>-W,,t^ (., 



Integrating between x and /, remembering that : 

dw 



dx 



for x = l: 



Hence 



^^S=t(^^-^')+&*'-^'^- • • • w 



^ '^(t^^.A ^1(^-1. 



-=£7lTly--^V+tU-^'^/'i- • • (3) 



The greatest deflection, w^, occurs for x = l. Hence 



I (WP pl<\ 



This value of w^ is the deflection of B below A. The 
general value of w in eq. (3) is the vertical distance (de- 
flection) of B below the point located by :^ ; as an ordinate 
it is measured upward from B as the origin of coordinates. 

The greatest moment, M^, exists at A, and its value is: 



M,^Wl + ^ (5) 



128 FLEXURE. [Ch. II. 

These equations are made applicable to a cantilever 
with a uniform load by simply making W =o. They then 
become 

^^=<-? (^) 



«^=;rl^r(^-^'*) (8) 



6£/' 






M,=\ do) 



Again, for a cantilever with a single weight only at its 
free end, p is to be made equal to zero in the first set of 
equations. Those equations then become : 



M^El'^^.^Wx , . (ii) 



dw W 



WP . . 

M,=Wl (15) 



Art. 28.] THE NEUTRAL CURVE FOR SPECIAL CASES. 



129 



The general expressions for the shear and the intensity 
of loading are : 



S-EI^=W+px, 






(16) 
(17) 



Case II. 

This case, shown in the figure, is that of a non -continu- 
ous beam, supported at each end, and carrying both a 



Fig. 2 

uniform load (whose intensity is p) and a single weight F/ 
at its middle point. The reaction R, at either end, will 
then be 

2 



R 



The general value of the moment will then be 



M=E1 



d^ 



R.-^-^ 



(18) 



The origin of x and w is taken at A, 
Remembering that 

dw . / 

-T-=o for %=--. 
dx 2 



130 FLEXURE. [Ch. II. 

and integrating between the limits x and -, 

Again integrating 



"^ EI 



I \R[x' xP\ p/x' xP\) , , 



The greatest deflection 7v^ occurs at the centre of the 
span, for which 



x=-. 

2 



Hence 



«''=-^/W + 8^^^ (^^) 



The greatest moment, also, is found by putting 



x=-. 

2 



It has the value 



M.=-i^'+f)' ..... (22) 



4\ 2 



These formulae are made applicable to a non-continuous 
beam carrying a uniform load only, by putting W = 6. 
They then become 

MORI'S -^^l-^h .... (23) 



Art. 2§.] THE UBUTkAL CUkvB FOR SpECML CASES. t^I 

EIt~= ( — ]» .... (24) 

ax 2 \ 2 3 12/ ^ ^^ 

P 
w^~^j{2xH-x'-Px), .... (25) 

Spl' 5 pi* 

'^''^ ~Js^^ ~s-4SE'r • • • • (26) 

pp 

M,=\ (27) 

The formulae for a beam of the same kind carrying a 
single weight at the centre are obtained by putting p =0 
in the first set of equations. Those for the greatest deflec- 
tion and greatest moment, only, however, will be given. 
They are 

WP 
^^=-^8£/' (^^) 

M, = ^ (29) 

The general values of the shear and intensity of loading 
are 

d'M , , 

1^=-^ (3^^ 



Case 111, 

The general treatment of continuous beams requires the 
use of the theorem of three moments. The particular case 
to be treated is shown in Fiij. %. The beam covers the 



132 . FLEXURE. [Ch. 11. 

three spans, DA, AB, and ^BC, and is continuous over the 
two points of support, A and B. 



Let DA =1^ 
'' AB=L 
- BC = l]i 



■ Let I2 = n/j = n'/g. 



Let the intensity of the uniform load on AB he repre- 
sented by p and let the two single forces P and P' only, act 




in the spans DA and BC respectively. Also let the two 
distances 

DE =z^= al^ and CF = a'l^ 

be given. // is required to find the magnitudes of the forces 
P and P\ if the beam is horizontal at A and B. 

Since the beam is horizontal at A and B, the bending 
moments over those two points of support will be equal 
to each other, for the load on AB is both uniform and 
symmetrical. Let this bending moment, common to A 
and B, be represented by M^. As the ends of the beam 
simply rest at D and C, the moments at those two points 
reduce to zero. 

Because the four points D, A, B, and C are in the same 
level, the first member of eq. (14) of Art. 24 becomes equal 
to zero. 

If that equation be applied to the three points D, A, 



Art. 28.] THE NEUTR/IL CURVE FOR SPECIAL CASES. 133 

and B, the conditions of the present problem produce the 
following results: 

. M,=o, M, = M^=M,. 



and 

llP(L'-z'')z = p 

4 



I ^ /' 

j~IP{l'-z''')z = p ' 



Hence the equation itself will become 

M,(2z.+3g+f(/.^-VK+/t=°- • • (32) 
'1 4 

•• ^^^'- 4^.(2/, + 3y ' 

2 1 4(2 + 3W) ^^^^ 

.-. Reaction at Z?=i?i=P^^ + ^l . . (34) 



As the origin of z^ is at D, x will be measured from the 
same point. 

Separate expressions for moments must be obtained for 
the two portions, DE and EA of /p because the law of 
loading in that span is not continuous. 

Taking moments about any point of EA 

EI^-R,x-P{z-z,) (35) 

Remembering that 

dw 



134 FLEXURE, [Ch. II. 

for x = l^, and integrating between the limits x and l^ 

dw R P 

EI^^--f(x'-h')--{x^-l,')+PB,(x-l,). . (36) 

Again, remembering that w=o for x=-l^, and integrat- 
ing between the limits x and /^ 



EIu;=-^[--l,^x + -^)--[~-l,^x + -f) 

+ P.,(^-/,^ + ^). (37) 



2 \3 ^ 3 / 2^3 



Taking moments about any point in DE 

E/jjj.K,«; (38) 

••• ='S=«.7+<^- <3<.' 

Making i\:=;Si in eqs. (36) and (39), then subtracting 

/. EI^-=^(x^-h')-j{z,^-h')+Pz,(z,-l,). (40) 

Remembering that w = o for x=o, and integrating be- 
tween the limits x and o, 

EIw=^{- -l,'x) -^(z,'-l,')x + Pz,{z,-l,)x. (41) 
Making r^ = 2i in eqs. (37) and (41), then subtracting 

i^ --(/.»- V)+^a/-2/)=o. . . (42) 

3 3 ^ 



Art. 28.] THE NEUTRAL CURVE FOR SPECIAL CASES, 13 S 

Putting the value of M^ from eq. {t^^) in eq. (34), then 
inserting the value of R^, thus obtained, in eq. (42), after 
making z^ =al^, 

L 2+sn ^ 2 ^ A 4(2+3^) 

• ' ^ 6a{i-a') 6aii-a'y ' ' ' ^^3; 

This is the desired value of P, which will cause the 
beam to be horizontal over the two points of support A 
and B when the span AB carries a uniform load of the 
intensity p. 

By the aid of eq. (43), eq. (^s) ^ow gives 



^{2n^ + Sn^) pn\^ pl^ 



2 -^ ^ 12(2 + 3^) 12 12 ^^^^ 

It is to be noticed that M^ is entirely independent of 
l^ or /g. Eq. (43) also gives 

^=^ n\ (45) 

Hence 

PI 
M, = —f{i-a')a. ..... (46) 

Thus any of the preceding equations may be expressed 
in terms of p or P. 
R^ also becomes 

pnl, _pnl^ 

^^~6a(i+a) ^7' W) 

or 

R,=P{i-a)[i-^a{i+a)] (48) 



136 FLEXURE. [Ch. II. 

It is clear that there cannot be a point of no bending in 
DE. Hence the point of contra-flexure must lie between 
E and A, Fig. 3. In order to locate this point, according 
to the principles already established, the second member 
of eq. (35) must be put equal to zero. Doing so and solving 
for X, 

P 

^=jrzR^^ (49) 

Since P is always greater than R^, there will always be 
a point of contra-flexure. 

All these equations will be made applicable to the span 
EC by simply writing a' for a, /g for Z^, and n' for n. 

As an example, let 

a=^ and n = i. 
Eqs. (43), (44), a^d (47) then give 



M.= 



P=Uh 
pP 3PI 



12 16 ' 

after writing, 

In general, the span l^ is called " a beam fixed at one 
end, simply supported at the other and loaded at any point 
with the single weight, P." 

Let it, again, be required to find an intensity, '' p' ,'' of a 
uniform load, resting on the span l^, which will cause the 
beam to he horizontal at the points A and E. 



Art. 28.] THE NEUTRAL CURVE FOR SPECIAL CASES, 137 

Since the load is continuous, only one set of equations 
will be required for the span. The equation of moments 
will be 

^^~dx^ =^1^- — (50) 

Integrating between the limits x and Z^, 

^^£4(^^-^>^)-f'(--^')- • • • (SO 

Integrating between the limits x and o, 

^^-f (f -'■')-?(t-'>)- • ■ <s=) 

But, also, w=o, when x--^l^. Hence 

R^-j^s' ■■■R.-iP'h (S3) 

This equation gives the value R^ when p' is known. 
Making x=l^ in eq. (50), and using the value of R^ from 
eq- (53). 

M, = pV{i-h) = -^-^ (54) 

Adapting eq. (32) to the present case, 

4(2 +3^) ^^^^ 

Equating these two values of M^, 

P'=ipn\ (56) 



138 FLEXURE. [Ch. II. 

Thus is found the desired value of p\ In this case the 
span /j is called " a beam fixed at one end, simply sup- 
ported at the other and uniformly loaded." 

The points of contra-fiexure are found by putting the 
second member of eq. (50) equal to zero and solving for 
X, after introducing the value of R^ from eq. (53). Hence 

ll^x — x'^=o, 
or 

x = o and x--^ll^. 

Between the simply supported end and point of contra- 
fiexure the beam is evidently convex downward, and convex 
upward in the other portion of the spans l^ and l^, whether 
the load is single or continuous. Moments of different 
signs will then be foimd in these two portions, and there 
will be a m^aximum for each sign. The location of the 
sections in which these greatest moments act may be made 
in the ordinary manner by the use of the differential cal- 
culus; but the negative maximum is evidently ilf,, given 
by eqs. (44) and (55). On the other hand, the positive 
maximum is clearly found at the point of application of 
P in the case of a single load, and at the point 

^ = 8 h 

in the case of a continuous load. These conclusions will at 
once be evident if it be remembered that the portion of the 
beam between the supported end and point of contra- 
fiexure is, in reality, a beam simply supported at each end. 
These moments will have the values 

i\/r TDi f N / AP{^ZL^^W±pan% 
M,=Pl,{i-a)a-i, ^(,_^3^) , . (57) 

M/=jhpV' (58) 



Art. 28.] THE NEUTRAL CUR^E FOR SPECIAL CASES. 139 

In case of a single load if P is given, and not p, eq. (45) 
shows 

M,=P/,(T-a)a[i--ia(i+a)]. 

The points of greatest deflection are found by putting 
the second members of eqs. (36), (40), and (51) each equal 
to zero, and then solving for x. They are not points of 
great importance, and the solutions will not be made. 

The following are the general values of the shears for a 
single load on /^ : 

InAE, S^E1^,=R,-P; [from eq. (35)]. 

In ED S,=EIj^,=R,; [from eq. (38)]. 

The shear in ^^ for the uniform load p^ is 



S'=-Elj^=R^-p'x\ [from eq. (50)]. 



Also 



Intensity of load = EI -r-^ = —p' - 

As has already been observed, all the equations relating 
to the span l^ may be made applicable to the span l^ by 
changing a to a' and nton'. 

The span l^ remains to be considered. 

Since the bending moments at A and B are equal to 
each other, and since the loading is uniformly continuous, 
half of it (the load pl^ will be supported at A and the other 
half at B. In other words, the vertical shear at an in- 
definitely short distance to the right of A, also to the left 



HO FLEXURE. [Ch. II. 

pi 
of B, will be equal to ^ Let x be measured to the right 

and from A. The bending moment at any section x will be 

^_(i'w ^. , pl^ px"" 



or 



EI^.-M, + ^{l,x-x^). . . . (59) 
Integrating between the limits x and o, 

Again, integrating between the same limits, 



''^'■P^'- I). ... (6x) 



Elw^—'^+^ikx'- 

2 12 



Since 

dw 

dx 

for x = l^, eq. (6o) wid give ii/2 independently of preceding 
equations. Following this method, therefore, 

2 12 

This is the same value which has already been obtained. 
Introducing the value of M^, 



duj_p/l^_x^_l,' 
dx 2\ 2 3 6 



Ei-£=i{^-^—H^ ■ ■ ■ (63) 



12 ^ ^ 



^■-i). , . . <... 



Art. 28.] THE NEUTRAL CURVE FOR SPECIAL CASES. 141 

The points of contra-fiexure are found by putting the 
second member of eq. (62) equal to zero. Hence 



l^\4-i 



x=lj-±\l-— -7 

^/ '0.211L. 



The moment at the centre of the span is found by 
putting 

L 



x = ^ 



in eq. (62): 

24 

This is the greatest positive moment 
The general value of the shear is 



^=='1:^ 



and the intensity of load 



The span /^ is generally called " a beam fixed at both 
ends and uniformly loaded." 

It is sometimes convenient to consider a single load at 
the centre of the span /,, while the beam remains horizontal 
at A and B\ in other words, to consider " a beam fixed at 
each end and supporting a weight at the centre." 

Let W represent this weight; then a half of it will be 
the shear at an indefinitely short distance to the right of 



142 FLEXURE. [Ch. II. 

A and left of B. As before, let x be measured from A, and 
positive to the right. The moment at any point will be 

£/g.M.-^. (65)* 

Integrating between x and o, 



EI-r~=M2X (66) 

ax 4 



If ^ = — , then will 

2 

dw _ ^ 
dx 

hence M2 = —z—. 

8 

The general value of the moment then becomes 

If X =- in this equation, the bending moment at the 
centre (where W is applied) has the value 

Centre moment = ——^. 

o 

Hence the bending moments at the centre and ends are each 
equal to the product of the load by one eighth the span, but 
have opposite signs. 

A second integration between x and o gives 

WKx' W 
12 

Hence the deflection at the centre has the value 



'"-Ei[--ir--^i (68) 



Centre deflection = f,,. 
' ig2EI 



* The use of the signs in this and the following equations is changed 
from the preceding to show that either procedure may be employed. 



Art. 28.] THE NEUTRAL CURVE FOR SPECIAL CASES. 



143 



By placing M =0, the points of contra-flexure are found 
at the distance from each end, 



X. = 



Addendum to Art. 28. 



The formulce of this article furnish the solutions of many 
practical questions of maxima deflections and moments. 
The latter for several ordinary cases are given in the follow- 
ing tabulation : 

P is the weight in pounds at end of beam or centre of span. 

p is the load in pounds per tin. ft. of beam. 



II 



III 



IV 



V 



VI 



VII 



vin 



Beam. 



Maximum 

Moment. 




Maximum 
Deflection. 



PI Sit A 



^pP at A 



\Pl at centre 



l/)/^ at centre 



■^^Pl at A 
-i^Pl at centre 



-ipPaiA 
+ ^13/^/2 at f/ 

from B 

-^Pl at A 
IPt at centre 



-^pl^ at A 
Ytpl"^ at centre 



I PP 



2i6|r>at A 



PP 
! 36 ^TT- at centre 

Pl' 

22. 5^^ at centre 



16 16^ at 0.447 



9-35 



p/3 



^* at 0.42 15/ 
EI from B 



Q-=ry^ at centre 
hi 



4.5^ at centre 



Point of 
Contra-flexure. 



j\lfTomB 
Reaction at B 



II from B 
Reaction at B 

-ipi 

\l from each end 



o. 2 1 1 / from each 
end 



144 FLEXURE. [Ch. II. 

I is the length of beam or of span in jeet. 

E is the coefficient of elasticity in pounds per sq. inch. 

/ is the moment of inertia of the normal section of the 
beam with all dimensions of section in inches. 

The " Max. Moments" will be in foot pounds, and the 
" Max. Deflections " will be in inches. 

In the use of flexure formulae, in many practical appli- 
cations, it is best to- have the moment M in inch-pounds, 
which will result from simply multiplying the " Max. 
Moments " of the preceding table by 12. 

Case I results from eqs. (14) and (15); Case II from 
eqs. (9) and (10); Case III from eqs. (28) and (29); 
Case IV from eqs. (26) and (27). In Case V the reaction 
is found by putting a = \ in eq. (48); the point of " Max. 
Deflection" is found by placing z^ =y in eq. (40), and the 

resulting value of -v- equal to zero and solving for x, which 

latter value in eq. (41) will give " Max. Deflection." 
Case VI results from treating eqs. (53), (51), and (52) in 
precisely the same manner. Case VII results directly 
from the formulae on page 142. Case VIII results directly 
from the equations on pages 140 and 141. 

The preceding cases are those which commonly occur 
with constant values of E and I. Other cases, such as a 
single load at any point, or partial uniform load over any 
part of span, are to be treated by the same general prin- 
ciples. 

Art. 29. — Direct Demonstration for Beam Fixed at One End 
and Simply Supported at the Other Under Uniform and 
Single Loads. 

A beam is said to be fixed at one end when it is under 
such constraint that the neutral surface does not change 
its direction at that end whatever may be the loading. 



Art. 29.] DEMONSTRATION FOR BEAM FIXED AT ONE END. 



45 



This fixedness, as has been fully shown in Art. 28, is equiv- 
alent to the application of a suitable constraining moment. 
Beams with one or both ends under such constraint have 
been fully treated in Art. 28, but it is desirable to establish 
the forniulae for such cases directly, i.e., without the employ- 
ment of the theorem of three m^oments. 

In Fig. I a beam is shown fixed at one end B and simply 
supported at the other end A, while it carries a uniform 
load p, per linear unit and the single load P at the distance 
at from A. The length of span is / and the coordinate x 




Fig. I. 



is measured horizontally to the right from A. The two 
reactions are R and R\ E is the modulus of elasticity, 
I the moment of inertia of the normal section of the beam, 
and w is the deflection at any point. The bending moment 
for any point in the segment al of the beam is : 



EI 






= Rx-.p— [^i) 



The bending moment for the section l—aloi the beam is 



EI^=Rx-p--P{x-al). 
ax"^ 2 



(2) 



Integrating eq. (i) and representing by C the constant 
of integration: 



dx 2 



P^+C. 



(3) 



146 FLEXURE. [Ch. 11. 

Integrating eq. (3) between x and 0, remembering that 
"0^=0, when x =o', 

EIw = R--^ + Cx (4) 

6 24 

Integrating eq. (2) between x and /, remembering that 

dw , , 

— - = o wnen x = l, 
ax 

EI^ = R"^-^ -|(^3 _/3) _|(^2 _/2 _ 2al{x-l)) . (5) 

If x=al in eqs. (3) and (5), the first members of those 
equations will be equal, hence: 

(ei'^^)=R^-P<!^+C . . . .■ (6) 
\ ax / 2 6 

(ei'^) =R-ia^ -i) -^{a^ -i)+^{a-iy . (7) 
\ ax / 2 6 2 

Taking the difference betw^een (6) and (7) and solving 
forC: 

C^-^+il+^'-ia-.y (8) 

2 2 

Placing this value of C in eq. (4) : 

EIw=~(x^-3l-x)—^(x^-4l^x)-^^a-iyx. . (9) 
6 24 2 

Integrating eq. (5) betw^een the limits of x and /: 

EIw =^(^3 -3/^+2/3) -J^(x^ -4/3^+3/4) 
24 

-^(x^-^3Px-\-2l^-3al(x-l)^) .... Cio) 
o 



Art. 29-] DEMONSTRATION FOR BEAM FIXED AT ONE END. 147 

Making x=al in eqs. (9) and (10) and subtracting the 
former from the latter, there will result : 

R=ipl+^ia^-sa + 2) (11) 

2 

This equation gives the reaction required to enable any 
of the preceding formulas to be applied to actual compu- 
tations. The loads P and p, as well as the quantity a are 
obviously known for any particular case or problem. With 
the value of the reaction now established by eq. (11) the 
deflection or the tangent of inclination of the neutral sur- 

face -7- may be at once com.puted for any point in either 
ax 

part of the beam. The fixing or constraining moment 

required to keep the beam horizontal at B can be at once 

determined by making x = l in eq. (2) and it has the value; 

M=Rl-^-Pl{i-a) (12) 



If the load is wholly uniform or P =0, eqs. (11) and (i) 
give: 

R=ipl and M=-^ (13) 



This value of M is the constraining moment required 
at B when the load is wholly uniform and is identical with 
eq. 54 of Art. 28. Indeed the preceding equations are the 
same as those established for the continuous span, con- 
ditioned similarly to the beam treated in this article. 

In all the preceding equations if the load is wholly 
uniform it is only necessary to make P =0. On the other 
hand, if there is a single load with no uniform loading 
p = o. 



148 FLEXURE. [Ch. 11. 

Inasmuch as the beam is convex downward over its 
left-hand part and convex upward in the vicinity of B, 
there must be a point of contrafiexure either to the right 
or to the left of P, according to its location. If that point 
is between P and B, the second member of eq. (2) must 
be placed equal to o, giving; 

^9{P-R) Pal ■ , s 

^'+-^— ^^ = 2-— (14) 

P P 

Solving this quadratic equation; 
R-P 



%=■ 



J2Pal/P-RY , . 



Eq. (15) gives the location of the point of contra- 
fiexure by the value of x measured from A. There are 
two roots of the equation, but evidently the positive value 
of the radical only is required. 

If the poi^t of contrafiexure is between P and A , which 
would be the case if the single load were near the right- 
hand end of the span, the second member of eq. (i) must 
be placed equal to o, giving; 

X=^ . (16) 

P 

In case the point of contrafiexure is at the point of 
application oi P, x= al, hence, 

2R T . 2R / N 

x= — = a/anaa= — - (17) 

P Pl 

If it is desired to find the point at which the deflection 

diju 
is geratet, it is osnly necessary to place — =0 in either 



Art. 29.] DEMONSTRATION FOR BEAM FIXED AT ONE END. 149 

eqs. (3) or (5), as the case may be, and solve the resulting 
eq. for x. 

The reaction R, i.e., the end shear at B, is; 

FJ=pl+P-R (18) 

The sum of the two reactions must be equal to the 
total load on the beam. 

Special Case, a=^. 

In this case eq. (11) will give the reaction i? at A as 
follows : 

R = lpl+^P (19) 

Hence, the reaction R' at B will be: 

R'=pl+P-R = ipl+iiP (20) 

The fixing or constraining moment Mi at B is by 
eq. (12): 

^--f-fa^^ (") 

Eq. (15) shows that the position of the point of contra- 
flexure will depend upon the magnitude of P. If P = o 
that equation shows that the point of contrafiexure will 
be |/ from A : 

% = f/ (22) 

The part f Z of the span will be in the condition of a 
beam simply supported at each end and uniformly loaded. 
Hence the greatest positive bending moment at the dis- 
tance f/ from A is: 

M'=^i^pP. ...... (23) 



I50 



FLEXURE. 



[Ch. II. 



The point of greatest deflection will be found by placing 
the second member of eq. (5) =0 and solving for x. 

Art. 30. — Direct Demonstration for Beams Fixed at Both Ends 
under Uniform and Single Loads. 

Fig. I shows a horizontal beam with both ends fixed, 
so that whatever may be the magnitudes of the uniform 
loading and the single load, or the position of the latter, 
the neutral surface at each end of the beam remains hori- 




FlG. I. 

zontal. The coordinate x is measured from the left-hand 
end A of the beam as is also the distance al of the single 
load P from the left end of the span. The length of the 
span is / and the reactions or shears at the ends of the 
span are indicated by R and R\ The fixing or constrain- 
ing moment at A is indicated by Mo and the uniform load 
per linear unit by ;^. If as before w represents the deflec- 
tion at any point, the equation of moments for the part 
al of the beam may at once be written : 



EIp^=Mo+Rx-p- 



(0 



Integrating eq. (i) between the limits x and o, and 



remembering that -7- = o for x=o] 
ax 

EI~=Mox+R- 
ax 2 



px^ 



(2) 



Art. 30.1 DEMONSTRATION FOR BEAMS FIXED AT BOTH ENDS. 151 

Integrating eq. (2) between the limits x and o, eq. (3) 
may be at once written as w=o for x=o: 

/yZ /Y*J ^Y^ 

EIw = Mo-+R^-p- (3) 

2 O 24 

Proceeding in the same manner for that part of the 
beam between B and the load P the equation of moments is : 

EIp^=Mo+Rx-p---P{x-al). . . (4) 
Integrating eq. (4) between the limits of x and /, since 

dw r 1 

-;— = o tor x=i\ 
dx 

EI ^ =Mo{x^-P) +-(^2 _/2) _|(^3 _/3) 

ax 26 

-^(x'-P-2alix-l)) (S) 

Again integrating between the limits x and I: 

EIw=^{x-l)^+^{x^-sPx + 2P)-£-{x^-4lH-^sl^) 
26 24 

--(--Px-alx^ + 2aPx+iP-aP\ , . (6) 

The two unknown quantities Mo and R are to be found. 
By placing x = al in eqs. (2) and (5), then subtracting the 
former from the latter : 

o=-Mo--/+^+-(a-i)2. ... (7) 
202 

Again making x=al in eqs. (3) and (4), then subtracting 
the former from the latter; 

o = (2a - 1) —ri^ci - 2) +^ (4^-3) H (a - 1)^. (8) 

2 6 24 3 



152 FLEXURE. '[Ch. II. 

If eq. (7) be multiplied by 4(2a — i) and then subtracted 
from eq. (8), the following value of the reaction or end 
shear will at once result : 

i?=-^^+P(a-i)2(2a + i) (9) 

2 

By placing this value of R in eq. (7), the value of Mq 
at once follows: 

Mo=-^-Pla{a-iY (10) 

12 

In order to determine the moment Mi at the end B 
of the span, it is only necessary to substitute the preceding 
values of Mo and i? in eq. (4) : 

Mi = -^-Pla?{i-a) (11) 

12 

These equations give all the quantities required for the 
complete solution of the case. The reaction or end shear 
at B is simply : 

^/+P-i? = ^^+P(i-(a-i)2(2a + i)) . . (12) 
2 

The greatest negative bending moment will obviously 
be found at either one end or the other of the span, depend- 
ing upon the value of a c.nd the amount of the load P. 
The greatest positive bending moment will be found where 
the shear is zero. 

There will be two points of contraflexure, one in each 
segment of the span. That point located in the part at 
will be determined in the usual manner by placing the 
second member of eq. (i) equal to zero and solving the 
quadratic equation. This simple operation will give eq. 

(13): 



R . hMo.R" 



,--^Vir+^ ^''^ 



Art. 30] DEMONSTRATION FOR BEAMS FIXED AT BOTH ENDS. 153 

Proceeding in the same manner with eq. (4) there will 
result : 

This last value of x will indicate the point of contra- 
fiexure for the right-hand part of the beam. 

Special Case, a=|. 

If P be placed at the center of the span, a = J and eqs. 
(9), (10), and (11) will give eq. (15): 

R=^+^ and Mo=Mi=-^-^. . . (15) 
22 12 8 

The moment M^ at the centre of the span will be given 
by the aid of eq. (i) : 

M'^il+q 06) 

24 8 

The greatest deflection wi is at the centre of the span 

and it is given by placing :^ =- in eq. (3). 

2 

The values of Mo and R are given by eq. (15). 



Art. 31. — Deflection Due to Shearing in Special Cases. 

The deflection due to transverse shearing only in all 
the ordinary cases of loaded beams can readily be com- 
puted by aid of the general eq. (9) of Art. 27. If ti is 
the distance from the most remote fibre from the neu- 



154 



PLEXVRE. 



tCh. ii. 



tral axis of any normal section whose moment of inertia 
about the same axis is I, and if G and 5 are the coefficient 
of elasticity and total transverse shear respectively, the 
deflection, w\ sought is 



w 



JIG. ' ^^•^- 



;/■ 



(i) 



The limits of the integration must be indicated for 
each particular case. 




Fig. I. 

In Fig. I let the cantilever, whose length is /, carry the 
single load P at its end, and the uniform load p per 
linear unit. The shear at any section distant x from .4 is 
S =P ^px. The substitution of this value of 5 in eq. (i) 
will give 



w = 



2 



. (2) 

and if P only acts. 



If the uniform load only acts, P =o 
p =o. 

Fig. 2 shows the case of a simple beam supported at 
each end, carrying a uniform load p per linear unit and 
the single load P at the centre of the span. The reaction 
R^^{P + pl), and the shear S=R — px. Hence eq. (i) 
gives the general value of the deflection 

px)dx=^-fy.\-{P-\-pl)-'-~\. (3) 



u/ = 



2lG 






Art. 31.] DEFLECTION DUE TO SHEARING IN SPECIAL CASES. I55 

pP 



And for the centre ot the span: 



/f2 /»//« J2 / 



(4) 




Fig. 2. 



The values of the deflection w' may be S-.milarly written 
for other cases. The following table gives the results for 
the cases indicated, which are those commonly required. 



II 



III 



IV 



VI 

VII 
VIII 



Beam. 




A 



'^^P per unit 
^^ of length. 




End Shear. 



t\P 
t\P 

ipi 
iP 
hpi 



Shear 5. 



px 

hP 

Pi^l-x) 



Pill-x) 

hp 
Pihl-X) 



Section for 
Deflection w'. 



from end 



.447^ 

.4215/ 

.5/ 

.5^ 



Deflection w' 



dm 
2IG 

(Ppp 
4IG 

8IG 

fpp 
16IG 

14.327G 

d'pi 
12.SIG 

.QTf^.jd'^pP 

' 7g 
dm 

bIG 

d'pp 
I6IG 



156 FLEXURE. [Ch. II. 

The end shears in this table are the reactions taken 
from the table of the preceding article, the "Beams" in 
the two tables being the same. 

The total deflection for any particular beam is to be 
found by adding the "Max. Deflection ". from the table 
of the preceding article to the w' found in the above table. 

In the notation of the preceding article, if w^ is the 
deflection due to the lengthening and shortening of the 
fibres the total deflection in any case will be 

w =w^-\-w' (5) 

These formulae for shearing deflection, like all the 
formulas relating to the distribution of transverse shearing 
in a bent beam, are more accurately applicable to rectan- 
gular or circular sections than to others. 



Art. 32. — The Common Theory of Flexure for a Beam Composed 
of Two Materials. 

The common theory of flexure as set forth in the pre- 
ceding articles is applicable to a beam composed of two 
or more materials with minor changes only in the formulas 
established, but two different materials only will be con- 
sidered here, as that number are frequently used in engi- 
neering works. 

Two such materials, concrete and steel, are widely used 
in reinforced concrete beams. Let E be the modulus of 
elasticity for steel and Ei for concrete, and let e represent 

the ratio between the two moduli, i.e., e=^^. This ratio 

for concrete and steel is generally taken as 15, although 
12 is sometimes used. Let A be the area of that part of 
the section with the modulus E and Ai, the area of section 



Art. 32.] THE COMMON THEORY OF FLEXURE. 157 

having the modulus Ei. li u=- (the reciprocal of the radius 

p 

of curvature) be the strain of a unit length of fibre at unit 
distance from the neutral axis, then will the intensities 
of the direct stresses of tension or compression at the dis- 
tance z from the neutral surface be: 

Ni=EiMZ and N =Euz = eE\uz. 

Inasmuch as the two materials are supposed to act together 
as a unit, the rate of strain will be the same for both at a 
given distance from the neutral axis. 

The amounts of direct stress on the two differential 
areas dA\ and dA will be as follows: 

EiuzdAi-\-EuzdA=a\zdA\-\-eaizdA. . . (i) 

ai and noi are intensities of stsess at unit distance from the 
nutral axis. 

The sum of the direct stresses of tension and compression 
in any normal section of the beam, if the beam is hori- 
zontal and all loading vertical, will be zero. Hence: 

fzdAi-\-fezdA=o (2) 

The limits of the integrations indicated will depend upon 
the form of cross-section and the distribution of the two 
materials. Frequently the section of one material, such 
as the steel in reinforced concrete work, is but a small per- 
centage of the total cross-section, and it is sufficiently 
accurate to consider it concentrated at the distance d2 from 
the neutral axis on one side of the latter and at the dis- 
tance ds from the same axis on the opposite side. If J2 
is considered positive, ds must be taken as negative. 



158 FLEXURE. [Ch. II. 

Finally, if ^2 and ^3 be taken as the small areas of section 
of the material, eq. (2) will take the form of eq. (3) : 

fzdAi-\-{A2d2-A3ds)=o (3) 

Invariably the small sections A 2 and A3 belong to a 
material with a far higher modulus than the other. In 
reinforced concrete the sum oi A2 and A3 is usually about 
I per cent or less of the entire cross-sectional area of the 
beam with £^=30,000,000 and Ei =2,000,000. 

When the form of cross-section of the beam, i.e., the 
cross-section of both materials of which the beam is com- 
posed, is known, the position of the neutral axis of the 
section can at once be found by either eq. (2) or eq. (3). 
It is obvious from these equations that the neutral axis 
will not pass through the centre of gravity of the section. 
Whether it will be at one side or the other of that point will 
depend upon the amount and distribution of the materials 
and the greater modulus of elasticity. 

Frequently the steel is omitted on the compression side 
of reinforced concrete beams and in such case either A 2 or 
Az will be zero. 

The bending or resisting moment of the internal stresses 
in any normal section of a beam can be written at once 
by the aid of the second member of eq. (i). If that second 
member be multiplied b}^ z, the differential resisting moment 
will at once result. Hence: 

M =aijz^dAi+eaifz^dA. .... (4) 

As indicated in eq. (i), ai is the intensity of the direct 
stress of either tension or compression in a fibre at unit 
distance from the neutral axis for the material with the 
modulus El. The integrals in eq. (4) will be recognized 



Art. 32.] THE COMMON THEORY OF FLEXURE. 159 

at once as the moments of inertia of the cross-section of 
the two different materials about the neutral axis established 
by eq. (2) or eq. (3). If the same assumptions made in 
connection with eq. (3) are known in connection with eq. 
4 this latter equation will take the following form : 

M=ai^z'dAi+eai{A2dS+A^dh)- . . (5) 

Again since cLi = j- = -r = -r ^Q- (s) ^^Y ^^^^ ^^^ follow- 
di d2 ds 



ing form : 



M=^I+J-^l2+e^h (6) 

di d2 dz 



It is to be observed that k, k2 and ks are intensities of 
stress at the distances from the neutral axis indicated by 
di, d2, and ds in the material whose modulus of elasticity 
is El. 

These equations indicate completely the only modi- 
fications to be made in the common theory of flexure as 
applied to one material for a beam composed of two differ- 
ent materials, and they indicate also the corresponding 
changes necessary to adapt the common theory of flexure 
to a beam composed of more than two different materials. 

In eqs. (4), (5), and (6) the moment M is simply the 
ordinary expression for the external bending moment to 
which a beam is subjected in terms bf the horizontal co- 
ordinate X and given loads. 

The formulae to be used to compute the deflection of 
a beam composed of two materials are readily written by 
means of the preceding equations. As 



k k2 ^ T- El r- d^W 

ai=—=— =etc. =Eiu = — = ^1-7^' 
di (22 p dx"^ 



eq. (6) gives: 



t6o flexure. [Ch. II. 

Ei{I+el2+eIs)^=M (7) 

As already explained, M, the external bending moment, 
is expressed in terms of the loads and the coordinate x. 
Eq. (7) therefore can be integrated precisely as in the case 
of a beam of a single material. Indeed there is no differ- 
ence between the two cases except that istead of the 
moment of inertia I for a single material, the term I -\-eI2 + 
elz must take its place, the latter expression being the sum 
of the three components of the resultant moment of inertia 
of the combined normal section. 

The first integration of eq. (7) will obviously give the 
tangent of the inclination of the neutral surface at any 
point, while the second will give the deflection. 

Art. 33. — Graphical Determination of the Resistance of a Beam. 

The graphical method is well adapted to the treatment 
of beams whose normal sections are limited either wholly 
or in part by irregular curves. In Fig. i is represented 
the normal section of such a beam, the centre of gravity 
of the section being situated at C. The lines //L, AB, 
and DF are parallel. As is known by the common theory 
of flexure, the neutral axis will pass through C. 

Let aa be any line on either side of AB, then draw the 
lines aa' normal to AB, having made MN and HL equidis- 
tant from AB. From the points a' thus determined draw 
straight lines to C. These last hnes will include intercepts, 
66, on the original hnes aa. Let every hnear element 
parallel to AB, on each side of C, be similarly treated. All 
the intercepts found in this manner will compose the shaded 
figure. 

This operation, in reality, and only, determines an 



Art. 33.] GRAPHICAL METHODS APPLIED TO BEAMS. 



16: 



amount of stress with a uniform intensity identical with 
that developed in the layer of fibres farthest from the 
neutral axis, and equal to the total bending stress existing 
in the section ; this latter stress, of course, having a varia- 
ble intensity. HL represents the layer of fibres farthest 
from the neutral surface, consequently MN was taken at 
the samiC distance from AB. Any other distance might 
have been taken, but the intensity of the uniform stress 




Fig. I 

would then have had a value equal to that which exists 
at that distance from the neutral axis. Again, a different 
intensity might have been chosen for the stress on each 
side of AB. It is most convenient, however, to use the 
greatest intensity in the section for the stress on both sides 
of the neutral axis ; this intensity, which is the modulus of 
rupture by bending, will be represented, as heretofore, by i^. 
Let c and c^ be the "centres of gravity of the two shaded 
figures. These centres can readily and accurately be found 
by cutting the figures out of stiff m anil! a paper and then 
balancing on a knife-edge. Let s represent the area of the 



1 62 FLEXURE. [Ch. II. 

shaded surface below AB, and 5' the area of that above 
AB. 

Because this is a case of pure bending, the stresses of 
tension must be equal to those of compression. Hence 

Ks=Ks\ or s-=s' (i) 

The moment of the compression stresses about AB 
will be 

KsXc'C, 

The moment of the tensile stresses about the same line 
will be 

KsXcC. 

Consequently the resisting moment of the whole section 
will bo 

M=Ks{c'C-VcC)^KsXcc' (2) 

Thus the total resisting moment is completely deter- 
mined. In vSome cases of irregular section the method 
becomes absolutely necessary. 

It is to be observed that the centre of gravity, c or c\ 
is at the same normal distance from AB as the centre of 
the actual stress on the samiC side of AB with c or c\ 

Art. 34. — Greatest Stresses at any Point in a Beam. 

Any beam under transverse loading is subjected to 
internal stresses determined by the Common Theory of 
Flexure, the intensities of fibre stresses varying directly as 
'the distance from the neutral axis while the transverse and 
longitudinal shears are distributed as indicated in Art. 00. 
The maximum intensities of the direct stresses and shears 
at any point, however, must be determined by the aid of 
the procedures given in Arts. 8 and 9. 



Art. 34-] 

The intensity of the direct tensile and compressive 
stresses in any normal section may readily be determined 
when the conditions of loading are known. The only 
stresses acting on any two transverse planes at right angles 
to each other, one horizontal and the. other vertical, are 
the direct fibre stress py and the longitudinal and trans- 
verse shear pxy. It is shown in Art. 8 that the two inten- 
sities of principal stresses are given by the following equa- 
tion for all points : 



p = hPy±^P^y^+\py' (l) 

Again, if a is the angle which the axis of X (vertical) 
makes with the direction of one of the principal stresses 
it is shown in the same article that 



tan 2a =-^^^ (2) 



By the use of these equations it is shown in Art. 10 
that at the neutral surface of the bent beam where the 
intensity of the transverse and longitudinal shear has its 
maximum value, i.e., f the mean intensity on the entire 
section, there will be two principal stresses of equal inten- 
sity, and of the same intensity as the shear, but of opposite 
kinds, one being tension and one compression, each making 
an angle of 45° with the neutral surface. This determines 
completely the state of stress at the neutral surface. In 
the same article it is shown that there is but one principal 
stress at the exterior surface and that is the ordinary fibre 
stress of flexure whose intensity is determined by the bend- 
ing moment at the normal section considered. This inten- 
sity may be called k. The greatest intensity of shearing 
stress at the surface of the beam where the intensity k 
exists is given by eq. (5) of Art. 9. One of the principal 



1 64 FLEXURE. [Ch. II. 

stresses, i.e., that one normal to the exterior surface of the 
beam will be zero. Hence the maximum shear will be 
found on two planes at right angles to each other and each 
at 45° to the surface of the beam, the intensity of the shear 
being one-half of the principal stress k. These consider- 
ations determine completely the greatest stresses at the 
neutral surface and at the exterior surface, upper or lower,, 
of the beam. There remain to be found the intensity of 
principal stress at all other points by means of eqs. (i) 
and (2). 

To illustrate the necessary procedures, let a steel beam 
of rectangular normal section be taken with an erfective 
span of 20 feet, and with a depth of 16 inches. For the 
purpose of these computations the beam may be consid- 
ered to have a lateral thickness or width of i inch, making 
the area of cross-section 16 square inches. The load per 
linear foot may be taken at 11 40 pounds, producing an 
extreme fibre stress of ^ = 16,000 pounds per square inch. 
If X be measured from one end of the span and if ±.z he 
measured upward and downward, respectively, from the 
neutral surface, the greatest value in either direction being 
8 inches, and if I be the moment of inertia of the normal 
section of the beam about its neutral axis, there may be 
written the following values for the bending moment and 
intensity of fibre stress at any distance z from the neutral 
axis, g being the load per unit of span : 

. Mz . 

M=^{l-x)x :. k=py=-^zx(l-x). . . (3) 
2 21 

The transverse shear at any section x from the end of 
the span is ' 



Art. 34.] GREATEST STRESSES AT ANY POINT IN A BEAM. 165 

It is found by eq. (6) of Art. 15 that the intensity of 
transverse and longitudinal shear at any point in a section 
of the beam is 



^'"'\'f -) (4. 



2/ 



The value of tan 20; giving the direction in which the 
principctl stresses act now becomes 



(l-2x)(--zA 



Fig. I shows a part of one-half of the beam under con- 
sideration, the effective span being 20 feet =240 inches. 
One end support is at B while CD is at the centre of the 
span. NN is a trace of the neutral surface. 

Normal sections of the beam were taken 2 feet apart 
at F, G, H and C and the directions and intensity of the 
principal stresses p were computed by means of eqs. (i) 
and (2) at four points 2 inches apart vertically, including 
the neutral surface and exterior surfaces at each of those 
sections. The curved lines drawn in Fig. i are each laid 
down in the direction of the principal stresses acting at 
each point, the curves having the plus sign representing 
the directions of principal tensile stresses, while those indi- 
cated by the minus sign show the directions of the prin- 
cipal compressive stresses at each point. Along the neutral 
surface NN all lines are inclined at an angle of 45° to that 
surface, while at each exterior surface one set of lines is 



i66 



FLEXURE, 



[Ch. II. 



parallel to that surface and the other at right angles to it. 
Wherever the curved lines cross they are at right angles to 





each other. The plus stresses at the upper surface of the 
beam and the minus stresses at the lower surface have 
zero intensities at those surfaces. At the centre of span 



Art. 34.] GREATEST STRESSES AT ANY POINT IN A BEAM. 167 

CD all lines are horizontal, as the shear at that point is 
zero. They are horizontal whatever may be the character 
of loading at the point where the bending moment is 
greatest, i.e., where the shear is zero. These curved lines 
representing the direction of the principal stresses at all 
points are sometimes called stress trajectories. 

Some important practical matters are based upon the 
existence of the principal stresses of tension and compres- 
sion at the neutral surface of a bent beam, those principal 
stresses making angles of 45° with that surface. In Fig. 2 
is shown a rolled I-beam, although this discussion is equally 
applicable to the web of a plate girder. 

Inasmuch as the inclined principal stresses of tension 
and compression act at the neutral surface, let that dis- 
tribution of principal stresses be supposed to exist through- 
out the entire web of the rolled beam. This condition may 
be represented by the sets of lines drawn in Fig. 2, each at 
an angle of 45° with the vertical line (or with a horizontal 
line). Let it be supposed that the entire web of the beam 
is composed of the strips shown, those indicated by the 
broken lines AB being subjected to tension in the left half 
of the beam and those represented by full lines, to compres- 
sion. Inasmuch as the strips AC will be subjected to com- 
pression they may approximately be considered columns 
with the length h sec 4S°=hV2. The thickness of the 
web of flanged beams such as plate girders is sometimes 
determined by an empirical formula based upon this long- 
column condition of stress. Any part AC of. the web is 
in fact not in a true long- column condition because the 
parts parallel to AB are in tension and tend to hold the 
parts AC in position. 

Again, it is sometimes supposed that the web of a flanged 
beam m.ay be considered approximately to be composed 
of a system of tension and compression web members like 



1 68 • FLEXURE. [Ch. II 

a truss represented by such sets of strips of metal SiS AB 
and AC. 

The condition of compression in which the web exists 
in the direction AC tends to buckle a thin web into corru- 
gations with their axes parallel to AB, and such girders 
exhibit that result when tested to destruction if the web 
is insufficiently stiffened. For this reason it has some- 
times been proposed to place the stiffeners on the webs 
of plate girders in the direction AC, Fig. 2, so as to 
prevent any buckling of the kind described. Such a 
method, however, is not satisfactory for a number of 
reasons. 

If the total transverse shear in any normal sections of 
the beam such as a vertical section through A or C be called 
5 then the average intensity of shear assumed uniformly 

5 
distributed over the section of the web would be s=—. 

th 

Since such a vertical section would cut the same number 
of inclined strips in tension and compression, the shear 

-V2 (sec. 45°) would be carried by each of the sets of 
2 

inclined strips whose normal section would be 

th cos 45 =-—=. 

V2 

Hence, the intensity of stress in each of the two sets of 
strips would be 

S /- th S 

-V2^— :.=— . 

- 2 V2 th 



This is the same intensity as the mean transverse shear on 
the section of the. web. According to this mode of treat- 



Art. 35.] THE FLEXURE OF LONG COLUMNS. 169 

ment, therefore, it is seen that the intensities of stress 
throughout the assumed 45° strips is the same as the inten- 
sity of the average transverse shear. This again is simply 
the condition which exists at the neutral surface of the 
solid beam as already found, except in that case the inten- 
sity of transverse shear at the neutral surface is one and 
one-half times the average intensity. 



Art. 35. — The Flexure of Long Columns. 

A " long column" is a piece of material whose length 
is a number of times its breadth or width, and which is 
subjected to a compressive force exerted in the direction of 
its length. Such a piece of material will not be strained 
or compressed directly back into itself, but will yield 
laterally as a whole, thus causing flexure. If the length 
of a long column is many times ^the width or breadth, the 
failure in consequence of flexure will take place while the 
pure compression is very small and neglected. 

As with beams, so with columns, the ends may be 
''fixed," so that the end surfaces do not change their 
position however great the compression or flexure. Such 
a column is frequently, perhaps usually, said to have 
fixed ends. If the ends of the column are free to turn 
in any direction, being simply supported, as flexure takes 
place, the column is said to have "round" ends. It is 
clear that if the column has freedom in one or several 
directions only, it will be a " round" end column in that 
one direction, or those several directions, only. It is 
also evident that a column may have one end round and 
one end flat or fixed. 

In Fig. I let there be represented a column with flat ends, 
vertical and originally straight. After external pressure is 




lyo FLEXURE. [Ch. II. 

imposed at A, the column will take a shape similar 
to that represented. Consequently the load P, at 
A, will act with a lever-arm at any section equal 
to the deflection of that section from its original 
position. Let y be the general value of that de- 
flection, and at B let y=y^. Let x be measured 
from A, as an origin, along the original axis of 
the column. In accordance with principles already 
established, the condition of fixedness at each ^ of 
the ends A and C is secured by the application of 
a negative moment, — il/. It is known from the 
general condition of the column that the curve 
of its axis will be convex toward the axis of x at 
^^^' ^' and near A, while it will be concave at and near 
B (the middle point of the column). Hence, since y is 
positive toward the left, and since the ordinate and its 
second derivative must have the same sign when the 
curve is convex toward the axis of the abscissae, the general 
equation of moments must be written as follows : 

d'^y 

^'i^-^'-py- •••••• (0 

Multiplying by 2dy, 

EI^^^=2Mdy~P2ydy; 
:.El[^y = 2Aly-Py' + ic=o);. . . (2) 

c=o, because the column has flat ends, and 

dy 
dx 

when y=o. Also 



Art. 35.] 


THE FLEXURE OF LONG COLUMNS. 




dy 


when y=yi; 






:. uJ'y^ 

2 



171 



(3) 



Eq. (2) now becomes 

\' y{y—y 

\lE . . 2y 
:. a;=\ — - versm"^--. 



liy=y,, 

I Iei 



(4) 



(5) 



In this equation / is the length of the coliimn. From 
eq. (5) there may be deduced 

P^^ (6) 

It is to be observed that P is wholly independent of the 
deflection, i.e., it remains the same, whatever may be the 
amount of deflection, after the column begins to bend. 
Consequently, if the elasticity of the material were per- 
fect, the weight P w^ould hold the column in any posi- 
tion in which it might be placed after bending begins. 
This result is for pure flexure, direct compression being 
neglected. 

Eq. (6) forms the basis of some old long column 
formtdae now out of use. It was first established by 
Euler. 



172 FLEXURE. [Ch. II. 

Some very important results follow from the conside- 
ration of Fig. I in connection with the preceding equa- 
tions. 

The bending moment at the centre, B, of the column 
is obtained by placing y=yi in eq. (i); its value is, con- 
sequently, 

M' = -Al+Py^=M (7) 

Hence the bending at the centre of the column is exactly 
the same {hut of opposite sign) as that at either end. Between 
A and B, then, there must be a point of contra-fiexure. 

Putting the second member of eq. (i) equal to zero, 
and introducing the value of M from eq. (3), 

Introducing this value of y in eq- (4), and bearing in 
mind eq. (5), 



n lEJ I 



(8) 



The points of contra-fiexure, then, are at H and D, 
II and |/ from A. 

Hence the middle half of the column (HD) is actually a 

column with round ends, and it is equal in resistance to a 

fixed-end column of double its length. 

/ 
Hence writing V for - and putting 2V for / in eq. (6), 



2 



^ = ~p- (9) 

Eq. (9) gives the value of P for a round-end column. 
Again, either the upper three quarters {AD) or the 
lower three quarters {CH) of the column is very nearly 



Art. 35] T^^ FLEXURE OF LONG COLUMNS. 173 

equivalent to a column with one end flat and one end round, 
and its resistance is equal to that of a fixed-end column 

whose length is - its own. Putting, therefore, 
o 





^^=ll. 


and introducing 






H'- 


in eq. (6), 






P = c 



D 12 ' (10) 



The last case is not quite accurate, because the ends of 
the columns HC and AD are not exactly in a vertical line. 

In reality, the column under compression may be com- 
posed of any number of such parts as HD, with the por- 
tions HA and CD at the ends, thus taking a serpentine 
shape, so far as pure equilibrium is concerned. In such 
a condition the column would be subjected to considerably 
less bending than in that shown in the figure. In ordinary 
experience, how^ever, the serpentine shape is impossible, 
because the slightest jar or tremor would cause the column 
to take the shape shown in Fig. i. Hence the latter case 
only has been considered. 

If r is the radius of gyration and 5 the area of normal 
section of the column, eqs. (6) and (9) will take the forms 

P ^Tz^Er^ , P 7:'Er' 

and 



5 P S P • 

Eq. (10) will, of course, take a corresponding form. 

P 
These equations evidently become inapplicable when ^ 



174 FLEXURE. [Ch. 11. 

approaches C, the ultimate compressive resistance of the 
material in short blocks. The corresponding values of (-] 



at the limit are 

E 



I IE . I 

-=27r\-^ and -=7r\ 



c ■ ■ ■ -(") 



for fixed and round ends respectively; other conditions of 
ends will be included between those two. 
If for structural steel 

£^=30,000,000 and C = 60,000, 

the above values become 140 and 70, nearly. 

Euler's formula, therefore, is strictly applicable only to 
structural steel columns, with ends fixed or rounded, for 
which l-^r greatly exceeds 140 and 70, respectively. 

If for cast iron 

-£ = 14,000,000 and C=^ 100,000, 
eqs. (11) give 

- = 74 and -=37, nearly. 

Euler's formula evidently becomes inapplicable con- 
siderably above the limits indicated, since columns in w^hich 

- has those values will not nearly sustain the intensity C. 

The analytical basis of " Gordon's Formula" for the 
resistance of long columns is so closely associated with the 
empirical that both will be treated together hereafter. 



Art. 36.] SPECIAL CASES OF FLEXURE OF LONG COLUMNS. 



175 



Art. 36. — Special Cases of Flexure of Long Columns. 

There are a few cases of flexure of columns which, 
while not frequently found in engineering experience, may 
be of some practical importance. The two or three which 
follow involve the integration of linear differential equations 
treated in advanced works on the integral calculus; con- 
sequently the operations of integration will not be given 
here, but the general integrals will be assumed. 



Flexure by Oblique Forces. 

In Fig. I let OA represent a column acted upon by the 
oblique force P, which makes the angle a with the axis of 
X. The column is supposed to be 
fixed in the direction of OX at 0, but 
the coordinates x and y are measured 
from the point of application A of the 
load P as shown in the figure. If 
right-hand moments are positive, and 
left-hand negative, the component 
P sin a will have the negative moment 
—P sin ax about any point 0'. The 
lever arm of P cos a, if the deflection 
y is positive, is +3/, and its moment 
—P cos ay is also negative. Hence 
the resultant moment of any force, P, 
in reference to the point 0' is 



M=EI 



(Py 



dx^ 
x — P cos ay 



— P sin a 



(i) 




Fig. I. 



For any number of forces or loads P there will obviously 
be a corresponding number of pairs of terms in the third 



176 FLEXURE. [Ch. IT. 

member of eq. (i). It will therefore be sufficient to treat 
one force P only. 

Eq. (i) may be put in the form, 

d^y ,9 P sin a . . 

j^+n^y= -^j-x=mx . „ , . (2) 

T ^1 . ^- o P COS a J P sin q; -r^ , . 

In this equation n^ = — =^ — and m = — z=^ — . bq. (2) 

EI EI 

may readily be integrated so as to give the following 

equation, Ci and C2 being constants of integration: 

y =^—Anx-\-Cismnx — C2Q'0snx} . • • (3) 

Using the values of m and n given above y may take 
the following form, observing that — = — tan a : 



^ . \P cos a ^, \P COS a ^ , . 

y=C smA/ — rt — 6 cosa/ — =- — ;t:— ^tana. (4) 



EI yi EI 

The coefficients C and C have the values 






C= — Ci tan 
and 

C = — C2 tan ax/TiT , 

N/PcOSo: 

and they may be treated as arbitrary constants to be 
determined by the conditions of each problem 

As X and the deflection y are measured from the point 
of application of the load P, if x =0 then must y =0. Hence 
by eq. (4), C =0. Consequently 



^ . JP cos a , . . 

= C smAi — ^77 — X— ^tano: .... (5) 



EI 



Art. 36.] SPECIAL CASES OF FLEXURE OF LONG COLUMNS. 177 

If a is greater than 90°, cos a will be negative and 
the exponential value of the sine may be used as follows: 

IP cos a 

Placing b =a/ — — — , and e being the base of the Naperian 
logarithms : 

y= — ^=(e^'''^^'-e-^''^~')-xtSina. . . (6) 

2V -I 

When cos a is negative bV — i is the square root of a 
C 
positive quantity, and will be rational. 

V —I 

Column Free at Upper End and Fixed Vertically at Lower 
End with either Inclined or Vertical Loading at Upper End. 

In this case the axis of x, Fig. i, is to be considered 

vertical with the column fixed at its base 0. In accord- 

dy 
ance with the latter condition -/- = at 0, i.e., when x = l = 

ax 

length of column. 
From eq. (5), 



dy r- Pcos a P cos a ^ , X 

^=CV-g^cos^-g^^-tana. . . (7) 

It is to be observed that P is not yet determined and 
that cosaI — X may vary largely (and periodically) 

while -p- remains unchanged. 
dx 

If the column carries a vertical load at its upper end 

a = o =tSin a, and when X =1,^=0. Eq. (7) then gives: 

ax 

Cos^j^l=o (8) 



1 78 FLEXURE. [Ch. 11. 

If / is any whole odd number from i to infinity, then 
there may be placed by the aid of eq. (8) : 

1=5 • ") 

If this value be substituted in eq. (7) after making 
a = tan a=o: 

dy ^ I P fir . . 

Eq. (10) shows that when x=—(f being any whole odd 

number) -r- =0. for cos - =cos 90° =0. 
ax 2 

Obviously P must have the smallest value which will 
satisfy eq. (9) ; but / cannot be smaller than i. Therefore 

p-^'i (") 

The carrying capacity of the column is thus seen to be 
independent of the deflection as was the case in Art. 35, 
but it must be observed that the effect of direct compression 
is neglected, i.e., it is a case of pure bending of excessively 
long columns. The end of the column considered here 
which carries -the vertical load is free to deflect laterally, 
whereas in Art. 35 both ends are supposed to be held 
against lateral movement. In the latter case the resist- 
ance is seen to be nine times as great as in the present. 

Eq. (11) can be found in a direct and simple manner 
by making M =0 in eq. (i) of Art. 34 and integrating the 
resulting equation. 



Since by eq. (8), cos^^—l=o, sin^-— , 



Art. 36.] SPECIAL CASES OF FLEXURE OF LONG COLUMNS. 179 

If therefore a=o and x = l in eq. (5), and if y is the 
deflection of the free end of the column in reference to 
the base, Fig. i, that equation will give: 

C=yi (12) 

Then 

y=yism^—x (13) 

For a given value of x, therefore, y varies directly as 
3'i and the relative deflections at the base and any point 
may be computed by the equation: 



yi \2/ / 



(14) 

Or in the ordinary case: 

-^=sm.— (15) 

y\ 2/ ^ ^^ 

It should be remembered that deflection is initiated by 
the load P determined by eq. (11) and that the deflection 
may take any subsequent value without increase of load. 

Problems for Chapter II. 

Problem i. — A beam simply supported at each end 
carries a load of 850 pounds per linear foot over a span of 
26 feet. Find the bending moment and transverse shears 
at the end and centre of span and at 2 points 3 feet and 1 1 
feet 6 inches respectively from the end. 

Ans. Moment at end is o; at 3 feet, 29,325 ft. -lbs. ; 
at II. 5 feet, 70,868.75 ft. -lbs. ; at centre, 71,825 
ft. -lbs. Shear at end is 11,050 lbs.; at 3 feet, 
8500 lbs.; at 1 1.5 feet, 1275 lbs. 



i8o FLEXURE. [Ch. II. 

Problem 2. — A beam or girder having a span length of 
41 feet carries a uniform load of 1200 pounds per linear foot 
and a single weight of 1800 pounds at the centre. Find 
the bending moments and the shears due to the uniform 
load and the single load separately at the ends and at the 
centre and at points 6 and 14 feet from the end. 

Problem 3. — In Problem 2 find the single weight which 
placed at the centre of the span will produce the same 
centre bending moment as the uniform load. 

Ans. 24,600 pounds. 

Again, find two weights placed 6 feet apart, i.e., one 
3 feet either way from the centre, which will produce the 
same centre bending moment as the uniform load. 

Ans. Each of the two weights is 14,406 pounds. 

Problem 4. — A beam or girder with a span length of 
31 feet carries a uniform load of 300 potinds per linear foot 
in addition to five loads, the first weighing 7000 pounds 
at a distance of 3 feet from the end ; the second weighing 
10,000 pounds 7 feet from the end; the third weighing 
11,000 pounds 14 feet from the end; the fourth weighing 
17,000 pounds 21 feet from the end, and the fifth weighing 
6400 pounds 27 feet from the end. 

Construct the shear and moment diagrams for this case, 
Fig. 2 of Art. 15 and Fig. 2 of Art. 12. 

Problem 5. — Find a uniform load for the same beam 
considered in Problem 4 which will have a centre bending 
moment equal to the greatest bending moment of that 
problem ; also another uniform load whose end shear 
shall equal the greatest of the two end shears of Prob- 
lem 4. Such uniform loads are called " equivalent uniform 
loads." 

Problem 6. — In Problem 2 the moment of inertia / 



Art. 36.J SPECIAL CASES OF FLEXURE OF LONG COLUMNS. i8i 

is 3570 (the unit being the inch), while £^ = 30,000,000, 
the beam being of steel. Find the tangent of inclination 
of the neutral surface at the end and at 10 feet from the 
end. Also find the deflection at the centre of span and 
at 10 feet from the end. Use eqs. (19), (20), and (21) 
of Art. 22. 

Partial Ans. Tangent 10 feet from end is .00344. 
The deflection at the same point is .53 inch. 

Problem 7. — In Problem 6 let it be required to ascer- 
tain how much additional deflection is produced by the 
transverse shear at the centre of the span and at 10 feet 
from the end. Let the coefficient of elasticity for shear 
{G) be taken at 12,000,000 pounds, while / = 357o and 
d=--i4 inches. 

Ans. Deflection at 10 feet is .0054 inch, and at the 
centre of span .0075 inch. 



CHAPTER III. 

TORSION. 

Art. 37. — Torsion in Equilibrium. 

The state of stress called torsion is produced when a 
straight bar of material, like a piece of round shafting, is 
twisted. Such a bar is represented in Fig. i, the axis of 
the piece being AB, and its normal cross-section having 
any shape whatever. In engineering practice the outline 
of that normal section is usually circular, although it is 
occasionally square. 




1-- 

FlG. I 




The twisting of the bar is done by the action of two 
equal and opposite couples acting in two planes, each nor- 
mal to the axis, but at any desired distance apart. The 
two couples are represented in Fig. i at each end of the 
piece in the two normal sections A and B. The forces 
and lever-arm of one couple are respectively P and e, and 



P' and e^ of the other. 



The moment of the first couple 

182 







a; 

> O 



^ 2 



.go 



5 -^ 

^ JJ cn 

■P3.g 

r— <1> O 



X 5 ^ 

u Si -2 



= a 






-s o 



Art. 37.) TORSION IN EQUILIBRIUM. 183 

will be Pe and that of the second couple P^e\ and if pure 

torsion is to be produced these two moments must be equal, 
but opposite to each other. Inasmuch as the moment of a 
couple is the product of the force by the lever-arm, the 
forces and lever-arms of the two twisting couples may vary 
to any extent as long as the moments remain unchanged. 

Although the system of forces to which a bar in torsion 
is subjected is such as to be in equilibrium, any portion 
of the piece will tend to have its normal sections like those 
at CD rotated over each other, the result being a small 
sliding motion arotmd the axis of the piece. Hence a 
torsive stress is wholly a shearing stress on normal sections 
of the piece subjected to torsion. It is further important 
to observe that inasmuch as a couple produces the same 
effect wherever it may act in its own plane, the actual 
twisting moment need not be applied with its forces sym- 
metrically disposed in reference to the axis of the piece; 
indeed, both of those forces may be anywhere on one side 
of the piece without varying the conditions of torsion or 
torsive stress to any extent whatever. 

It is known from the general theory of stress m a solid 
body that although there can be no stresses of tension and 
compression parallel to the axis of a bar under torsion, or 
at right angles to it, there will be such stresses of varying 
intensities on oblique planes. Inasmuch as the result of 
torsion is to slide normal sections each past its neighbor, 
the elastic torsive shear Hke any other shear will not change 
the volume of the body. The principal shearing strains 
will produce deformation without changing the dimensions 
whose X)roduct gives the volume. 

The exact and complete mathematical theory of tor- 
sion deduced from the general equations of equilibrium of 
stresses in an elastic solid, without extraneous assump- 
tions, will be found in App. I. Those formulas show accu- 



r84 TORSION. [Ch. III. 

rately the state of torsive stress in bars of any elastic 
material and of various shapes of cross-section. For the 
general purposes of engineering practice that general demon- 
stration is rather complicated. Hence it is often avoided 
by making certain approximate assumptions based to some 
extent on experimental observations which lead to an 
approximate and simpler theory, yielding formulas accurate 
only for the circular normal section, but which are not 
materially in error for the square section. These formulae 
are, how^ever, far from accurate for certain other sections. 
In this article only the formulae of the simpler theory, 
called the common theory of torsion, 
will be given. 

Fig. 2 is supposed to represent the 
normal section of a bar of material of 
any shape, subjected, to torsion by the 
application of couples as shown in 
Fig. I. The fundamental assump- 
tions of the common theory of tor- ^^^- ^■ 
sion are that the intensity of shearing stress varies directly as 
the distance from a central point at which that intensity is 
zero, and that that central point is located at the centre of 
gravity or the centroid of the section. It is also implicitly 
assumed that the normal sections which are plane before 
torsion remain plane during torsion. In Fig. 2, A is sup- 
posed to be the centre of gravity of the section at which the 
intensity of shear, i.e., the shear per square unit of section, 
is zero. The distance from the centre A to any point of 
the section is represented by r, and to the mxost remote 
point in the perimeter of the section by r^. In accord- 
ance with the assumed law, the greatest intensity of shear 
Tm in the section will be found at the distance r^ from its 
centre. While this is accurately true for the circular sec- 
tion, it is quite erroneous for a number of other sections. 




Art. 37.] TORSION IN EQUILIBRIUM. 1:85 

Hence the intensity at the distance unity from the centre 
A will be -7", and at the distance r from the centre it 
will have the value 



s = ~Tm (i) 

The element of the section at the distance r from A will be 

rdoj.dr (2) 

Hence the shear on that element is 

r Tm 

dS = — T m-'rdiij.dr=- -^r^dr.dco. ... (3) 
^0 ^0 - 

The direction of action of this torsive shear is around 
the circumference of a circle whose radius is r; hence if 
moments of all these small shears, dSy be taken about the 
centre or point of no shear, A, the lever-arm of each small 
force, dS, will be r, and the differential moment will be 

dM=rdS = —r'dr.daj. . . . . (4) 

The total moment of torsion therefore will be 

M = f^^ r ^r'dr.dco = ^ r r\'drdco= ^L. (5) 

Jo 7o r^ Tq Jo Jo Tq ^ ^^^ 

The quantity Ip is the polar moment of inertia of the section. 
For a circular section 

^ TIT,' Tzd' ^, ^. ^ Ad"" 

/^ = -^ == -- ( J = diameter) = -g-. . . (6) 



1 86 TORSION. [Ch. III. 

For a square section (6 = side of square) 

^ b' Ab' 

^^ = 6=^ (7) 

For a rectangular section {b = one side and c = the other 
side) 

, bc' + b'c A(b' + c') 

Ip = =-^ (8) 

^ 12 12 • • V^/ 

For an elliptical section (a^ and 6^ being semi-axes) 

;r(a,^^ + aA^) ;raA(a.H^^) 
/, = =— ^ . . . (9) 

Using the notation of Fig. i, the following equation of 
moments may be written, Pe being the moment of the 
external twisting couple and M the moment of the internal 
torsive shearing stresses in any normal section: 

Pe = M=^J, (lo) 

It is clear from Art. 2, if ^o is the shearing strain at the 
distance r^ from the centre, that T^,= (70 q, G being the 
coefficient of elasticity for shearing. Also, since the inten- 
sity of shearing varies directly as the distance from the 
centre A, it is equally clear that the shearing strain ^ 
varies directly as the distance from the centre, so that 
if a represents the shearing strain at unit's distance from A 

<j>=ra and ^0=^0^ (11) 

Hence in general 

T ==Gra, (12) 

and as a maximum 

Tm=Gr^a (13) 



Art. 37.1 TORSION IN EQUILIBRIUM, 1S7 

a is evidently the angle through which one end of a fibre 
of unit '5 length and at unit '5 distance from the centre or axis 
is turned. It is called the angle* of torsion. 

If / is the length of the piece twisted, the total angle 
through which the end of the fibre at unit's distance from 
the axis will be turned is 

Total angle of torsion = a,.. . . . (14) 

If the fibre is at the distance r^ from the axis one end 
will be twisted around beyond the other by an amoimt 
equal to 

Total strain of torsion =rf^aL . . . (15) 

By the aid of eq. (13) eq. (5) may be written 

Pe=M=GaI,=^I, (i6) 

If (f)Q is observed experimentally 

^=^ ^''^ 

The angle through which a shaft will be twisted by the 
moment Pe, if the length is /, is 

If 6* is in pounds per square inch, as is usual, the pre- 
ceding formulae require all dimensions to be in inches, 
while a will be arc distance at radius of one inch. 

If 12/ is written for / the unit for the latter quantity 
must be the foot. 

By inserting the value of l^ from eq. (6) in eq. (5), 

* This small angle is measured in radians. Strictly speaking it is an 
indefinitely short arc with unit radians. 



1 88 TORSION. [Ch. III. 



Pe = M= "^^ 



2Tm 7:d^ r.Tmd 






<i ■ 32 16 ' 
:. d = i.j.2\JY~~ (19) 



. , Eq. -(19) will give the diameter of a shaft capable of 
resisting the twisting moment represented by Pe with the 
maximum torsive shear in the extreme fibres of Tm- 
:. . The main cross dimensions of other sections may be 
found similarly by the use of eqs. (7), (8), and (9). 

It . is frequently convenient to compute the greatest 
intensity T m from the twisting moment M. For this pur- 
pose the equation preceding eq. (19) gives 

M ' 
Tm = S-^-^ (20) 

These equations complete all that are required for the 
practical use of the common theory of torsion. In some 
cases it may be necessary to use accurate formula for 
other shapes of section than the circular. In those cases 
the exact formula of App. I should be employed. The 
practical applications of the preceding formula to such 

Twisting Moment in Terms of Horse-power H. 

It is sometimes convenient to express the twisting 
moment M in terms of horse-power transmitted by the 
snafting. If H is the number of horse-powers transmitted 
by a shaft making n revolutions per minute, the inch-pounds 
of work will be 12 X33,oooXii^, since each horse-power repre- 
sents 33,000 foot-pounds of work performed per minute. 
Again if e is the lever arm of the twisting couple, the path 
of the force P per minute will be 2Ten and the work per- 
formed by the couple must therefore be PX2Tren=M2Trn. 
Equating these two expressions for the work or energy 
transmitted ; 



Art. 37-] TORSION IN EQUILIBRIUM. 189 

^^^^ =63,025— =M. . . . (21) 

27rn n 

If this value of M be placed in eqs. (19) and (20), the 
values of d, the diameter of the shaft and r,„, the greatest 
intensity of shear will take the following forms in terms 
of the horse-power and the number of revolutions per 
minute : 

'^ = ^^•^7^ (-) 



Hollow Circular Cylinders. 

If the exterior diameter of a hollow cylinder is d and the 
interior diameter di =jd, j being simply the ratio between 
the two diameters, the equation preceding eq. (19) may be 
written : 

M = '^{d^-d,^) .(24) 

Hence 

j^^p^^^J^nij^zfld^ . .-. . (25) 

16 ^ ^^ 

Eq. (25) shows that any of the preceding equations 
may be made applicable to a hollow cylinder by writing 
TmCi -f) in the place of 7^. 

Eqs. (19) and (22) therefore take the following forrns 
for a hoUow cylinder : 



The resistance of the hollow cylinder is obviously the 
difference between the resistances of two solid cylinders; 



I90 TORSION. [Ch. III. 

one having the exterior diameter and the other the interior 
diameter of the hollow cylinder. 



Art. 38. — Practical Applications of Formulae for Torsion. 

There has been comparatively little experimental inves- 
tigation in the resistance of structural materials to torsion 
and practically none of that has been done in connection 
with pieces of considerable size. Such results as have been 
obtained appear to justify the following data. 

Steel. 

Some of the older tests, as those of Kirkaldy, indicate 
that the ultimate intensity of torsional shear, T^, may be 
taken as high as 75,000 pounds to 90,000 pounds per 
square inch for special grades of steel like those used for 
tires, rails, and crucible steel, but lower values must be 
employed for mild structural steel and for the ordinary 
grades of shafting. 

Torsion tests on circular pieces of spring and cold-drawn 
steel about f inch and ij inches in diameter made in the 
testing laboratory of the Dept. of Civil Engineering at 
Columbia University by Mr. J. S. Macgregor gave the 
following results, which are show^n rather fully in order to 
exhibit clearly their main features. There were either four 
or six tests in each group from which the '' max.," " mean " 
and " min." were taken. All these test specimens except 
those of mild steel were heat treated. Part of these were 
heated to 1350° F. and then plunged in oil at 70° until cold. 
They were then temper drawn in hot oil at 575° F. and 
part were again heated to 1350° F. and immersed in oil at 
575° F. They were then allowed to cool in air at normal 
temperature. 



Art. 38.] APPLICATIONS OF FORMULAE FOR TORSION. 



191 





Diam. 
Inches. 


Pe. 

In.-Lbs. 


Elastic. 


Ult. 


Modulus. 
G. 


Spring steel 


max. 

■ mean 

min. 


.617 

■:6i4 


5.640 
5.427 
5.290 


41,600 
40,710 
31,050 


122,310 
118,110 
114,620 


13,010,000 
12,455,000 
11,292,000 


Spring steel 


max. 
• mean 
(^ min. 


1.252 
1 .246 


45,260 
44.040 
42,720 


46,100 
43.130 
41,500 


117,320 
115,070 
111,900 


13.954.500 
'12,659,000 
11,830,000 


Cold-drawn steel. 


max. 

■ mean 

min. 


1.252 

1-25 


43.500 
37.990 
34.270 


46,200 
39.300 
33,000 


113,200 
99,000 
89,500 


12,445,000 
11,602,000 
10,534,000 


Mild steel 


' max. 
mean 
min. 


1-257 
1-233 


24,500 
23,200 
21,520 


22,000 
20,800 
19,700 


62,800 
60,950 
57.300 


12,600,000 
12,110,000 
11,700,000 



It has been shown in Art. 5 that r = 



2G 



Hence 



if £=30,000,000, which is essentially correct for steel, and 
if 6^ = 12,000,000 as the mean, approximately, of the values 
in the preceding table, then wiir 



r = .25 

Direct torsion tests of six small nickel steel specimens 
by Prof. E. L. Hancock and described by him in Vol. VI 
(1906) of Proceedings of the American Society for Testing 
Materials gave elastic limits : 

Max. Mean. Min. 

Nickel Steel . .36,000 32,900 30,500 pounds per sq. in. 

He also found for mild carbon steel the two following 
elastic limits : -- 

Mild Carbon Steel. . . . 29,000. . . . 2 5, 500 pounds per sq. in. 

As the ultimate resistance of mild carbon steel to torsive 
or ordinary shear may be taken at about three-quarters 



192 TORSION, [Ch. Ill 

the ultimate tensile resistance, and approximately the same 
ratio between the elastic limits, it is reasonable to take the 
elastic limit in torsion at 25,000 pounds to 28,000 pounds 
per square inch for that grade of material having an ulti- 
mate tensile resistance of 60,000 pounds to 68,000 pounds 
per square inch. 

Nickel steel has a higher ratio of the elastic limit divided 
by the ultimate, and a mean value of 33,000 pounds per 
square inch for the elastic limit is reasonable. 

If the greatest intensity of torsive shear Tm allowed in 
the design of a shaft of diameter d is fTe in which Te is 
the elastic limit and/ a suitable fraction, perhaps .5 in some 
cases, then eq. (19) of the preceding article will take the 
form: 



^'\wrv-^^'^Y^ ^'^ 



Similarly eq. (22) of the same Art. will become: 

''=»-^=(-"^5l, ■■■<■) 

Wrought Iron, 

Wrought iron is now seldom used for shafting or similar 
purposes, but such tests as have been made show that the 
torsive elastic limit of wrought iron may be taken from 
20,000 pounds to 25,000 pounds per square inch and used 
as indicated in eqs. (i) and (2). From 10 per cent to 20 
per cent higher values may be taken for cold-rolled shafting. 

Cast Iron. 

Cast iron is ill adapted to resist torsion and is not 
commonly used for that purpose, yet it has been tested 



Art. 38.] APPLICATIONS OF FORMULA FOR TORSION. 193 

in torsion, although generally in special grades such as were 
formerly employed in making cannon or car wheels. Such 
grades of cast iron gave ultimate values of Tm from 24,000 
pounds to 45,000 pounds per square inch or even more, 
but they are far too high for ordinary castings used in 
engineering practice. Probably half the preceding values 
would be large enough for the best quality of ordinary 
castings, although the highly variable and erratic qualities 
of cast iron make it exceedingly difficult to assign exact 
data for purposes of design. The modulus of elasticity, G, 
may be taken at 7,000,000 for ordinary grades of cast iron, 
or at 6,000,000 for the lower grades. 

Alloys of Copper, Tin, Zinc and Aluminum. 

The torsional resistance of this class of alloys varies 
greatly with the relative proportions of their constituent 
elements in a manner quite similar to that exhibited by 
the corresponding resistance to tension. 

Professor R. H. Thurston was probably the earliest 
thorough investigator of the torsional resistances of many 
of these alloys. He found the ultimate intensity of torsive 
stress Tm to vary from a few hundred pounds per square 
inch to nearly 48,000 pounds per square inch for alloys of 
copper and tin running by gradual variation from pure cop- 
per to 10% of that metal alloy to 90% of tin. The alloy 
80-90% Cu with 20-10% vSn gave Tm varying from about 
47,700 pounds to 43,900 pounds per square inch with a maxi- 
mum twist of 1 14. 5 degrees. Similarly he found the ulti- 
mate Tm for pure copper to range from 28,400 to 35,900 
pounds per square inch with a total twist of over 150 de- 
grees. On the other hand, pure tin gave the ultimate 
T„»=32oo pounds (nearly), the total angle of twist running 
as high as 691 degrees. The elastic limit of the more due- 



194 



TORSION. 



[Ch. III. 



tile of these alloys was found to vary from about 35% to 
60% of the ultimate T^. The alloys running from 70% 
Cu with 30% Sn to 29% Cu with 71% Sn were brittle, 
giving low values of Tm from about 700 pounds per square 
inch to less than 6000 pounds per square inch; those alloys 
failed at the elastic limit with a total angle of twist of only 
I to 2 degrees. 

Similar results with like erratic variations were found by 
Professor Thurston for alloys of copper and zinc. The 
greatest values of Tm ran from about 35,000 to 52,000 
pounds per sq. in. for 90.58% Cu with 9.42% Zn to 49.66% 
Cu with 50.14% Zn. 

It should be observed that the test specimens used by 
Prof. Thurston were .625 inch, in diameter with a torsion 
length of I inch only and they were tested in his torsion 
machine. 



Table I. 
ALUMINUM ALLOYS— TORSIONAL RESISTANCE. 



1 

r ^ ^.n--r. u^.- r<=^+ Angle of ■ Torsive Shear 
Composition Per Cent. ; torsion Deg. : per Sq.In. 


General Character. 


Al. 


Sn. Cu. 


Elastic 
Limit. 


Maxi- 
mum. 


Elastic 
Limit. 


Maxi- 
mum. 


2.5 
2.75 
5 

6.25 
7.5 
8.75 
10 
*ii 

*20 


■ 2.5 
3.75 
5 

6.25 
7.5 
8.75 

10 

II 

20 


100 
95 

92.5 
90 

87.5 
85 

82. 5 
80 
78 
60 


2 

4 

6 

7 

4 

3.5 

7 

6 

5.8 

I 


130 
200 
198 
175 
37 

22 
10 

8 
5.8 

I 


4,300 
10,710 
11,827 
15,525 
30,282 
25,447 
18,413 
15,230 
13,717 

2,321 


25,000 
33,075 
35,802 
45.155 
63,440 
37,062 
18,413 
15,230 
13,717 
2,321 


Very soft; ductile. 

Soft; ductile. 

Slightly tough; ductile. 

Tough; medium ductility. 

Very tough; rather hard. 

Hard; somewhat brittle. 

Very hard; brittle. 

Very hard; exceed'gly brittle 

Very hard; cx.ceed'gly brittle 



2 

10 

12 
13 
85 
27 
100 



Scattering. 



10 


88 


3 


147.5 


14,000 


43,987 


I 


89 


5 


52 


21,740 


50,000 


2 


86 


9 


9 


32,984 


32,984 


2 


85 


8 


12 


32,723 


37,003 


7.5 


7.5 


3 


37 


8,703 


17,630 


119 


69.6 


2.5 


20 


2,800 


2,800 






2 


160 


4,005 


12. 911 



Somewhat soft; ductile. 

Tough; medium ductility. 

Very tough; hard. 

Very tough; hard. 

Very soft; somewhat ductile. 

Very soft; spong^'-. 



♦Could not be machined, 



Art. 38.] APPLICATIONS OF FORMVLJE FOR TORSION. 



195 



Table I contains experimental values of the elastic 
limit and ultimate torsion shearing resistance of the alloys 
of aluminum, tin, and copper shown in the table. They 
were determined by Messrs. Gebhardt and Ward in the 
mechanical laboratory of Sibley College at Cornell Uni- 
versity and reported to the Am. Soc. Mech. Engrs. in 
1898. 

The results of the table show that the alloys yielding 
other resistances of considerable value will also exhibit 
proportionate torsion resistances, as might be anticipated. 

The Eighth Report to the Alloys Research Committee 
of the Institution of Mechanical Engineers of Great Britain 
by Prof. H. C. H. Carpenter, M.A., Ph.D., and Mr. C. A. 
Edwards in 1907 contains some interesting torsion tests on 
specimens of copper-aluminum, the pieces being .624 inch 
in diameter and 3 inches in length with the exception of 
No. 3, which was 2.8 inches in length. Table II gives the 
results of these tests. It will be observed that alloys with 
a comparatively small percentage of aluminum give much 
higher torsional ductility than pure copper. This is proba- 
bly due to the fact that rolled copper generally contains 



Table II. 







Greatest Twisting Moment 






Cu. 


Al. 
Per cent. 


and Stress. 


Twist on 

Whole Length 

Degrees. 


Ratio, 
Torsion {T^) 

Tension 


Per cent. 


Moment. 


Stress, Lbs. 






In.-Lbs. 


per Sq.In. 






99 96 





1,792 


37,500 


2,736 


I 51 


99 


9 


. I 


2,293 


47,960 


5.184 




15 


98 


94 


1.06 


2,359 


49,350 


4.345 




41 


97 


9 


2 . I 


2,464 


^I'i^"" 


3,600 




34 


95 


95 


405 


2,813 


58,870 


2,316 


I 


2 


93 


23 


6.73 


3,306 


69,170 


1,623 




18 


92 


61 


7-35 


3.373 


70,580 


1.374 




15 


90 


06 


9-9 


3.351 


70,110 


234 





89 


88 


2 


11.72 


3,584 


74,970 


51 


I 


04 



196 TORSION. [Ch. m 

some dissolved oxygen which diminishes its ductiUty. The 
addition of a small amount of aluminum removes the oxy- 
gen and enhances the ductility. The authors of the report 
express the conclusion that " Alloys containing aluminum 
up to 7 J per cent behave extremely well under the torsion 
test but beyond this percentage there is a rapid deteriora- 
tion of properties." The ratio between the ultimate resist- 
ance Tm to torsional shear and the ultimate tensile resist- 
ance is shown in the last column of the table. 

Other Sections than Circular. 

The common theory of torsion is correct only for cir- 
cular sections. The general demonstration for other sec- 
tions than circular shows that for square, rectangular, 
triangular and elliptical sections, the maximum intensity 
of^torsive stress Tm will be found at the middle point of a 
side of a square section or of the longest side of a rectangular 
section, or at the middle point of the side of an equilateral 
triangular section and at the extremities of the minor axis 
of an elliptical section. If, however, for approximate pur- 
poses the formulae of the common theory of torsion should 
be used for the sections indicated above the polar . moments 
of inertia Ip would be taken from eqs. (6), (7), (8) and (9) 
of Art. 37. The maximum torsive shear Tm, in this pro- 
cedure, should be taken as existing at the extreme points 
of the section. The results by this approximate method 
will be sufficiently near for most ordinary purposes, at least 
with the square section, but the exact theory should be 
used for oblong sections or where the highest degree of 
accuracy is desired for non-circular sections. 



CHAPTER IV. 
HOLLOW CYLINDERS AND SPHERES 

Art. 39. — Thin Hollow Cylinders and Spheres in Tension. 

If a straight closed hollow cylinder be subjected to an 
interior pressure having the intensity q' sufficiently greater 
than that of the exterior pressure gi, there will be a ten- 
dency to split the cylinder longitudinally. 

Fig. I represents such a cylinder with sides so thin that 
the stress to which they are subjected may be considered 
uniformly distributed throughout 
any diametral section. If a cy- 
lindrical shell has much thickness 
relatively to its interior radius 
the tensile annular stress due to ~ 
inner pressure will not be uni- 
formly distributed throughout 
the shell. The excess of inner 
pressure over the outer, if the Fig. i. 

latter exists, will cause the inside 

part of the annular section of metal to be stressed to a 
higher intensity than the outside and that difference will be 
greater as the thickness of the shell increases relatively to 
the radius. It becomes necessary therefore to distinguish 
between these two classes of cylindrical shells in their ana-- 
lytic treatment. 

AB represents the diametral plane through the axis of 
the cylinder, the thickness i of the shell being supposed in 
this case to be so small that the cylindrical shell may be 
considered *' thin." 

197* 




198 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

As the notation shows r' is the interior radius and ri 
the exterior radius. If C, the centre of the cyHndrical 
section, be taken as the origin of the circular coordinates 
/ and a, and if a unit length of cylinder be considered, 
the indefinitely small amount of pressure on a differential 
of the interior surface /a will be q'/da and it will have a 
component at right angles to the diametral plane AB 
expressed by q'r'da sin a. The integral of this expres- 
sion between 180° and o will be the total normal pressure 
acting on the two longitudinal sections of metal at A and 
B, as shown by the following equation: 



I q'r' sin ada = 2q'r'. 



One-half of the second member of this equation, q'r\ 
represents the tendency to split the cylinder at either A 
or B and it must be resisted by the sections of metal at 
those two points, or at any other two points at the extrem- 
ities of a diameter. 

Precisely the same integration made for the exterior 
pressure will obviously give the quantity qiVi representing 
the tendency to give the metal compression at the extremi- 
ties of any diameter. 

The resultant tendency to split the cylinder per unit of 
length will then be q'r' —qiVi, it being supposed that the 
interior pressure is so much greater than the exterior that 
tension only will be induced in the material. Obviously 
if the exterior pressure were much larger than the interior, 
compression would exist instead of tension. The intensity 
of tensile stress t in the sides of the cylinder will therefore be 

. . i=^^^:^^:;^ (I) 



Art. 39.] THIN HOLLOW CYLINDERS AND SPHERES IN TENSION. 199 

This value of t expresses the tendency of the cylinder 
to split along a diametral plane under the action of the 
interior pressure q'. 

If the ends of the cylinder are closed, the internal 
pressure against them will tend to force them off or to pull 
the cyhnder apart around a section normal to the axis. 
The force F tending to produce this result will be 

F = r:iq'r''-q^r,^) (2) 

The area of normal section of the cylinder will be 
7:(r^- — /-). Hence the intensity of stress developed by 
this force will be 

If the exterior pressure is so small that it may be con- 
sidered zero, eqs. (i) and (3) give 

^ = V' • ^4) 

/=rfi72 (5) 

'1 ' 

When the thickness of the shell is small / may be 
ced ec 
will give 



r -\-r 
placed equal to — ' — ^, and this value introduced in eq. (5) 



<ir^ <lX (6, 

' 2(r, — r) 21 

f in eq. (6) is seen to be but half as much as t in eq. (4). 
In this case, therefore, if the material has the same ulti- 
mate resistance in both directions, the cylinder will fail 
longitudinally when the interior intensity is only half 
great enough to produce transverse rupture. 

In designing thin cylinders it will usually be necessary 
to determine the thickness i. so that the tensile stress t in 



200 HOLLOW CYLINDERS AND SPHERES, [Ch. IV. 

the metal shall not exceed the prescribed value h. After 
writing h for / in eq. (i), also r^ — / for i, then dividing 
both sides of the equation by r', there will result 



This equation readily gives 



,fh + q" 



If the exterior pressure q^ is so small that it may be 
considered zero, the thickness given by eq. (7) takes the 
following form : 

^-v w 

This is the same value that will be found by solving 
eq. (4) for i. 

The expression for the thickness of the material of the 
cylinder to resist the longitudinal tension having the in- 
tensity / can be foimd with equal ease. If f^ be written 
for / in eq. (3), as the greatest permissible longitudinal 
tension, then if both numerator and denominator of the 
second member of that equation be divided by /^ there 
will result 

The solution of this equation at once gives the desired 
thickness : 



^'/:t.j-' (') 



Art. 39.] THIN HOLLOW CYLINDERS AND SPHERES IN TENSION. 201 



If q^ is so small that it may be neglected, it is simply to 
be made zero in eq. (9). 

If the exterior pressure q^ were considerably larger than 
q^, the resulting stresses in the sides of the cyHnder would 
be compression, but the formulas for the resulting intensi- 
ties would be precisely the same as the preceding, as long 
as the cylinder retained its circular shape. 

The case of stresses in a thin hollow sphere or thin 
spherical shell may be treated in the same general manner. 
The hemispherical ends of a metallic cylindrical tank or 
reservoir may be illustrated by the skeleton section in 
Fig. 2. 





Fig. 2. 
As indicated in the figure the internal radius of each 
end is r', while r 1 is the external radius. The internal and 
external intensities of pressure are as shown in Fig. i. 
The force tending to tear off the hemispherical ends of the 
tank along the line AB, Fig. 2, is 7r(q/^ —qri^). The sec- 
tion of metal resisting this force with the intensity t is 
Tr(ri^—r'^). The intensity of stress developed in the metal 
will therefore be 



_q'r'^-qiri^ 



rr 



^f2 



(10) 



If the external pressure is so small that there may be 
taken gi =0, eq. (10) will take the form 



/=■ 



ri 



q'r'^ 



2i ' 



(11) 



202 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

In this last equation i=ri—r\ and the interior radius 
is placed equal to one-half the sum of the interior and ex- 
terior radii, as may be done without sensible error. The 
interior radius being given, the thickness of metal required 
to withstand a given internal pressure q' without stressing 
the metal above a given working value t may be written 
as follows from eq. (ii) : 



./../ 



*=17 • • ('^) 

If the value of the thickness i should be desired in 
terms of both the interior and exterior pressures, it can 
easily be written by the aid of eq. (lo) ; if both numerator 
and denominator of the second member of that equation 
be divided by r'^, there may at once be found 

ri (t+q' 



r \t-\-qi 

After multiplying this equation through by r\ then sub- 
tracting that quantity from each side of the resulting 
equation, the desired value of the thickness will be 

By giving a proper working value to the tensile in- 
tensity t and inserting the values of the pressures, the thick- 
ness i will at once result. 

In all these equations no allowance is made for the 
metal taken out by the rivet holes in riveted work. This 
does not, however, affect in any way the equations found. 
It is only necessary to remember that the cross-section of 
metal required by the preceding equations is to be regarded 
as the net section, i.e;, the section remaining after the rivet 



Art. 40. 



THICK HOLLOW CYLINDERS. 



203 



holes have been made. This is equivalent to making the 
thickness i great enough to give the required section as 
net section. 

Art. 40. Thick Hollow Cylinders. 

If the thickness of sides or walls of hollow cylinders 
and spheres subjected to high internal pressures is great 
in comparison with the internal radius, the tensile stress 
in the metal may not be assumed to be uniformly distrib- 
uted, and it is necessary to deter- 
mine entirely different formulas 
from those established in the pre- 
ceding article. 

The normal section of a thick 
hollow cylinder is shown in Fig. 
I , r' being the internal radius and 
ri the external, with the intens- 
ities of internal and external 
pressures p' and pi respectively. 
It is supposed that the internal 
pressure so greatly exceeds the 
external that the metal sustains 
tensile stress only. If the 
cylinder be supposed to be 

divided into a great number of thin concentric portions, 
the elastic stretching of the metal will cause a much higher 
tension to exist, in the interior portions than in the exterior. 
If any diametral section, such as AB, Fig. 1, be assumed, 
it is clear that the sum of all the tensile stresses developed 
in that section must be equal to the excess of the internal 
pressure over the external. A unit length of cylinder will 
be considered in the following formulae. 

The tensile stress in the sides of the cylinder, whose 
intensity will be represented by h, and which is developed 




204 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

in any diametral section, as AB, has a circumferential 
direction, and for that reason it is sometimes called " hoop 
tension." 

The variation of this tensile intensity h carries with it 
a corresponding variation in intensity of the radial pres- 
sure whose intensity is p, having the values p' in the in- 
terior of the cylinder and pi at the external surface. 

The amount of tension on a radial section of thickness 
dr will be hdr, and if that differential expression be inte- 
grated so as to extend over the entire thickness of one wall 
or side of the cylinder, it must be equal to the effort of the 
internal pressure in excess of the external to split the 
cylinder along one of its sides. The following equation 
is the analytical expression of this condition: 



pr-piTi 



^\dr (i) 



If p\ pi, r\ and ri be considered variable so as to refer 
to any interior points in the wall of the cylinder, and if r' 
and ri become so nearly equal to each other that ri—r' 
may be considered as aV, then will p'r' —piri=d{pr) and 
eq. (i) will become: 

d(pr) =pdr-\-rdp=hdr. ... (2) 

Eqs. (i) and (2) will be in no way changed if the ends 
of the cylinder are closed, it being assumed in that case 
that the longitudinal stress is uniformly distributed over a 
normal section like that shown in Fig., i . 

Eq. (2) is a differential equation expressing a relation 
between the two intensities p and h. Another equation of 
condition is required in order to determine the two unknown 
quantities. This second equation can be written by ex- 
pressing the relation existing between the direct and lateral 



Art. 40.] THICK HOLLOW CYLINDERS. 205 

strains due to the stresses p and /^, so as to leave the radial 
longitudinal sections of the walls of the cylinder plane under 
the conditions of stress due to the assumed internal and 
external pressures. The establishment of such an equation, 
however, will, lead to, or express, precisely the same con- 
ditions involved in the analysis of Art. 5 of Appendix I, 
which therefore need not be repeated here. Those con- 
ditions may be expressed by stating that the sum of the two 
intensities p and h, i.e., (p+h), is a constant for given 
intensities of pressure. If therefore, a be such a constant 
there will be assumed the equation : 

P+h = a (3) 

.*. dp= —dh, Sind p=a—h. 

By the aid of these expressions eq. (2) will take the 
form: 

2hdr-\-rdh=adr. 

By multiplying both sides of this equation by r there 
will result : 

d{r%)=-dr^ (4) 

2 

If 6 is a constant the integration of eq. (4) will give 



Also 



'=-A^ (5) 



2 r^ V / 



The interior and exterior pressures p^ and pi are known, 
and eq. (6) will give the two equations : 

P= 7^and^i= (7) 

2 r^ 2 Ti^ 



2o6 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

By subtracting pi from p' 

b=^^^,r'W (8) 

Then by the second of eqs. (7) : 

The substitution of these values of h and - in eqs. (5) 

2 

and (6) will give the following values of the intensities p 

and k. Inasmuch as the preceding equations involving h 

and p have been written without giving distinctive signs 

to either tension or compression and as the constants h 

and - m^ay be regarded either as positive or negative, the 
2 

sign of each one will be changed by writing ri^—r''^ for 

r'2— ri^, which will make the tensile stress k positive and 

the compressive stress p negative after substituting the 

values of the constants h and - in eqs. (5) and (6). 

2 

p'r'^-pin^ p'-pi /^n^ , . 

Eqs. (10) and (11) can be put in more convenient form 
for use in numerical computations by dividing both numer- 
ator and denominator of all the terms in the second mem- 
bers of those equations by ri^. This simple operation will 
give eqs. (12) and (13): . 



Art. 40] THICK HOLLOW CYLINDERS. 207 



P= z^- — zf¥- :^.' " • • (12) 





p'-p,r'^ 




/2 

p'%^-p^ 


• • 


y'2 


r'2 r2- • 





(13) 



Eqs. (12) and (13) are the general values of the inten- 
sities of the internal stresses in the walls of the cylinder, 
p acting in a radial direction and h in sl circumferential 
direction. The greatest tensile intensity h' will exist at 
the interior surface of the cylinder and it will be found 
by making f =f' in eq. (13) as shown by eq. (14) : 

'"^ 

Similarly the intensity of tensile stress at the outer 
surface of the cyhnder (the least intensity of tensile stress) 
will be given by making r =ri. 

r'2 / r 

^P'-.-p\^+:,^. 



The thickness t of the wall of the cylinder which must 
be provided if the greatest intensity of tensile stress h' is not 
to be exceeded by a given intensity p' of interior pressure, 



2o8 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

can readily be found by solving eq. (14) for the quantity 
— ^, which will give eq. (16). 



r' \2pi 



p'-^W, 



(16) 



Then by adding ( — 1) to each side of eq. (16) and 
multiplying both members by r' eq. (17) will at once 
result : 



n—r =t = r 






As the internal radius / will always be known, eq. (17) 
gives the thickness t desired in terms of the known pres- 
sures and the intensity of working stress h\ 

Eq. (17) shows that if 2pi+h' =p', t will be infinity. 
This shows that when the intensity of the internal pressure 
is equal to or greater than twice the intensity of the exter- 
nal pressure added to the greatest allowed tensile stress in 
the metal, it is impossible to make the wall of the cylin- 
der thick enough to resist that internal pressure. 

If the external pressure is so small that it may be 
neglected, it is necessary only to place pi=o in the pre- 
ceding equations. 

If pi exceeds p' it is obvious that the internal stress h 
will be compression, i.e., there will be hoop compression 
as the circumferential stress in the cylinder wall instead 
of hoop tension. 

The complete solution of the problem of the thick 
cylinder including expressions for the distortions or strains 
of the material at all points will be found in Art. 5 of 
Appendix I. 

The application of the preceding formulas can be ex- 
pedited by the use of the following tabular values which 



Art. 40.] 



THICK HOLLOW CYLINDERS. 



209 



explain themselves. A curve more useful than the table 
can readily be constructed from the numerical values in 
the latter, so that any value whatever for the ratio of the 
radii indicated can be read at sight. 



r' 


r'2 


r' 


r'2 


r 


n2 


r 


ri2 


I. I 




5 1 


25 




95 


9025 


45 


2025 




9 


81 


4 


16 




85 


7225 


35 


II25 




8 


64 


3 


09 




75 


5625 


25 


0625 




7 


49 


2 


04 • 




65 


4225 


15 


0225 




6 


36 


I 


01 




55 


3025 


05 


0025 



As an illustration of the laws of variation of the inten- 
sities h and p the following data may be used : 

f' = 10 inches; 

h' = 20,000 pounds per square inch; 

p' = 10,000 pounds per square inch; 

pi =1000 pounds per square inch. 

Eq. (17) will give, after a substitution in it of the above 
numerical data, ^ = 5.81 inches. 

fi = 15.81 inches. 



The quantity — will have values ranging from unity 



J2 



for the interior of the cylinder to — =.4. 

ri 



Inserting these 



values in eqs. (12) and (13) there will result the two 
equations : 



p = 5000 — 1 5 ,000 — ; 



2IO HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

h = SOOO + 15,000 ^r. 

Taking the varying values of — ^ given in the above 
table the following values of p and h will result : 



• r' 
r 


Pounds per Sq.in. 


P » 


r 


95 
9 

85 
8 

75 
7 

65 
63 


- 10,000 20,000 

- 8,540 18,540 

- 7,150 17,150 

- 5,840 15,840 

- 4,600 14,600 

- 3.440 13,440 

- 2,350 12,350 \ 

- 1,340 11,340 j 

- 1,000 11,000 j 



Fig. 2 represents these results graphically. The straight 
line GAP is laid off tangent at any point A to the circle 

HB D 




Fig. 2. 



representing the interior of the cylinder subjected to the 
pressure of 10,000 piounds per square inch. Similarly HBD 



Art. 40.] THICK HOLLOW CYLINDERS. 211 

is a straight line laid off tangent at the point B on any radius 
CB of the exterior surface of the cylinder, the distance AB 
being equal to 5.81 inches. AF is then laid off by scale 
equal to h' = 20,000 pounds, while AG is similarly laid off to 
represent p' = 10,000 pounds per square inch, but it must be 
remembered that it acts in a radial direction, i.e., along 
AB. BD and BH are the corresponding quantities for the 
exterior surface of the cylinder, equal respectively to 11,000 
pounds and 1000 pounds. Curves DF and HG are then 
constructed by laying off the ordinates p and h at right 
angles to A B as shown. 

Gase of Exterior Pressure Greater than Interior Pressure. 

If the exterior pressure pi is greater than the interior 
pressure p' , it is evident that the preceding equations will 
need no change whatever, but the difference p' —pi will 

now be negative. As p'^ is less than p' , p will still be 

negative and represent compression. On the other hand, 
h will now be negative and represent circumferential or 
hoop compression as shown by eq. (13). Eqs. (12) and 
(13) are used in connection with this case in designing 
modem heavy guns where thick cylinders are raised to a 
high temperature and slipped over a close-fitting interior 
thick cylinder at ordinary temperature, so that when the 
outside hot cylinder cools it contracts and puts the interior 
cylinder under a high compression. In fact, the lining of 
the gun may be enclosed by two or more such cylinders 
successively shrunk into place. One interior cylinder with 
slightly conical interior surface may be forced by a high 
pressure at ordinary temperature into the interior of a 
corresponding exterior cylinder with similar results. These 
matters will be treated more extendedly in the next article. 



212 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

Art. 41. — Radial Strain or Displacement in Thick Hollow Cylin- 
ders. — Stresses Due to Shrinkage of One Hollow Cylinder 
on Another. 

Radial Strain or Displacement. 

Inasmuch as all diametral sections of thick hollow 
cylinders remain plane for all conditions of stress due to 
internal or external pressure, the only strain or displace- 
ment in such a cylinder is that in a radial direction due to 
either increase or decrease of the diameter of any elementary 
thin cylinder or shell with radius r. This radial displace- 
ment will be indicated by p and the expression for it can 
only be established by the analysis shown in Art. 5 of 
Appendix I, or by some equivalent analysis. By referring 
to eq. (10) and the two equations preceding eq. (15) of that 
article it will be seen that the desired displacement is given 
by the following equation : 

{i-2r)[pi-p'—)r-{-{p,-p')- 

^ ''' -^. . . (i) 



J2 
2GI- 



r'2 \ 



In this equation G is the modulus of elasticity for shear- 
ing, while pi and fi represent the intensity of exterior 
pressure and the exterior radius, respectively, and p' and 
r' similar quantities for the interior of the cylinder. The 
quantity r represents the ratio of the lateral strain divided 
by the direct strain, i.e., Poisson's ratio. Obviously if 
r=r\ the increase or decrease of the interior radius will be 
given by p and a similar observation applies to the increase 
or decrease of the exterior radius when r = f 1. 

It is clear that if r be made equal to either r' or ri in 
eq. (i) either p' or pi may be written from that equation 



Art. 41] STRESSES DUE TO SHRINKAGE. 213 

in terms of the corresponding radial displacement p' or pi. 
It is sometimes desirable to express the intensities of 
interior or exterior pressures in this manner, after having 
determined the radial displacement corresponding to a 
known- change of temperature or in some other manner. 
In the operation of shrinking one cylinder on another the 
difference in diameters required for the operation may be 
prescribed by some empirical rule. 

Stresses Due to Shrinkage. 

It has been shown in the preceding article that when a 
thick hollow cylinder is subjected to a high internal pressure 
the intensity of circumferential or hoop tension is much 
greater at and near the interior surface of the cylinder than 
at the exterior surface and that if the thickness is great 
the intensity of the interior tension may be high, while 
that of the exterior surface will be extremely low, show- 
ing the use of the metal to be uneconomical. In heavy 
gun making this undesirable condition is overcome by 
dividing the body of the gun into a number of concentric 
thick cylinders, each being shrunk over those inside of it, 
after making the interior diameter at ordinary temperature 
less than the exterior diameter of that over which it is 
shrunk into place. Each tube is heated so as to enlarge 
its diameter until it can be slipped over the tube, or tubes, 
inside of it, so that when it cools it will itself be subjected 
to high internal pressure with correspondingly high cir- 
cumferential or hoop tension, while the tube, or tubes, 
inside of it will be correspondingly compressed at ordinary 
temperature. The body of the gun thus composed of a 
series of concentric tubes shrunk in place in series will form 
a combination in the interior of which there will be rela- 
tively high circumferential or hoop compression, decreasing 



214 



HOLLOW CYLINDERS AND SPHERES. 



[Ch. IV. 



outwardly though not regularly or continuously, with cir- 
cumferential or hoop tension in the outer part or parts. 
When the intensely high pressures of modern explosives 
are produced in firing the gun the metal will be more nearly 
uniformly stressed in circumferential tension and thus act 
more effectively throughout the entire thickness of the 
wall of the gun. It will not be attempted here to give 




Fig. 1. 



the details required to secure the best effects by shrinking 
into place a series of thick hollow cylinders in the manu- 
facture of ordnance, but the general analytic procedure 
in deducing the proper results for the shrinkage of one 
cylinder on another either in gun making or in the making 
of other compound cylinders for high internal pressures 
will be illustrated by a single computation only. 

Fig. I represents a thick hollow steel cylinder with 



Art. 41.] STRESSES DUE TO SHRINKAGE. 215 

internal diameter of 12 inches and total thickness of wall 
of 12 inches composed of an outer cylinder 6 inches thick 
shrunk on an interior hollow steel cylinder with wall also 
6 inches thick. It will be supposed that the coefficient 
of expansion of steel per degree Fahr. is 8 = .0000065. The 
increase in diameter due to a change of 225° Fahr. of the 
interior 24-inch cylinder will be 225 X 24 X 5 = .0351 inch. 
The change in radius will be one-half of this amount. The 
interior diameter of the exterior thick cylinder at ordinary 
temperature must be 24 — .03 51 =23.9649 inches. 

If /' be the interior radius of the exterior cylinder 
before being heated and r^^ the exterior radius also before 
being heated, while / and ri represent the interior and 
exterior radii of the interior cylinder at ordinary tempera- 
ture and before shrinkage, as shown in Fig. i, the data 
required will be as follows : 

r'=6''; ri = i2''; r'^ = 11.98245 ; r^, = 17.98245. 

The interior pressure of the inner cylinder will be simply 
that due to atmosphere. Similarly the exterior pressure 
on the exterior cylinder will also be that due to the atmos- 
phere. Hence, both these pressures will be considered 
zero. There will then be acting the shrinkage pressure on 
the exterior surface of the inner cylinder and the same 
pressure on the interior surface of the outer cylinder. The 
intensity of this common shrinkage pressure will be indi- 
cated by ^1. 

As indicated in Fig. i, after the properly heated outer 
cylinder has been slipped over the inner cylinder at ordi- 
nary temperature and the two allowed to cool, the radius 
ri will be decreased by the radial displacement pi, while 
the radius /' will be increased by the amount p'\ In- 



2l6 



HOLLOW CYLINDERS AND SPHERES. 



[Ch. IV. 



asmuch as pi will be intrinsically negative, eqs. (2) will at 
once result. 



ri + pi=r +p 



P"-Pi=ri-/'. 



(2) 



By making p' =0 in eq. (i) and r=ri there will result 
eq. (3): 



Pi = 



f(i-2-)ri+- 



2G 



rr 



(3) 



Similarly by making pi=o in eq. (i), r' =r'\ ri=r^^, 
r = r" and remembering that p' =p" =pi, eq. (4) will repre- 
sent the increase of the radius of the exterior cylinder after 
the operation of shrinkage is complete : 



// __\ rn / 

'2 X 



(4) 



By substituting the second members of eqs. (3) and (4) 
for the first member of eq. (2) an equation will result 
giving the value of pi as shown by the following equation 
and eq. (5) : 



.'fs 



2G 
Or 

Hence 



(i-2-r)\J^-{-^ 






f'2 










n^ 




1 


r" 




r'2 




1 


y"2 




ri^ 


i 




ru' 


1 



= Y\ —Y 



P^f^-n-r". 



pl = - 



2G{n -r") 



(5) 



Art. 41.] STRESSES DUE TO SHRINKAGE. 217 

The quantity represented by Z \s clear. 

It should be observed that the relation between the 
changes of the radii at the cylindrical surface of shrinkage 
contact and the original radii (eq. (2)) is general and holds 
for all conditions of shrinkage stresses whatever may be 
the thicknesses, of the two cylindrical walls. 

In computing the value of pi by eq. (5) there will be 
taken : 

6^ = 12,000,000 and r=.2 5. 

If the values already given for the four radii of the cyhnders 

be inserted in the equation immediately following eq. (4), 

there will result : 

^=-38.4. 
Hence 

.0351 X.5 X24,000,000 1, . , 

pi =—^^ n "= 10,970 lbs. per square mch. 

38.4 

In computing the value of Z it is essentially accurate 
to use the inner and outer radii of the outer cylinder as 
they exist before it is heated for the shrinkage process. 
This will save much labor and simplify the application of 
the formulas, but the difference ri—/' must of course be 
expressed as accurately as possible. 

The stresses in the walls of the two cylinders due to 
shrinkage may now be readily computed, since the outer 
cylinder is subjected to an inner pressure of 10,970 lbs. per 
square inch and the inner cylinder to an exterior pressure 
of the same intensity. The resulting values of p and h 
for the two cylinders are as follows : 

Inner Cylinder in Compression. 

^1 = 10,970 lbs. per square inch; p' =o] r' =6 inches; 
fi = 12 inches. 



2l8 



HOLLOW CYLINDERS AND SPHERES. 



[Ch. IV. 



Eqs. (12) and (13) of the preceding article will give at 
once eqs. (6) and (7) for this case: 



/2 



p=pl 



(6) 



ri' 



iH — -o 



h=p] 



(7) 



— I 



Tl' 



If the intensities p and k are computed at six equidistant 
points at the two surfaces and at intermediate points one- 
fifth of the thickness of the wall of the cylinder apart, the 
results given in the following tabulation will be found and 
they are shown graphically in Fig. 2. 



r 


y'2 

r2~ 


Pounds per Sq. In. 


V' 


■ i 






P 


h 


I 


I. 


i —29.250 


I .2 


.6944 


- 4.470 


-24.780 


1-4 


.5102 


- 7,164 


— 22,090 


1.6 


.3906 


- 8,913 


-20,340 


1.8 


.3086 


-10,113 


— 19,089 


2 


•25 


-10,970 


-18,283 



Outer Cylinder in Tension. 

^' = 10,970 lbs. per square inch; p\=o\ r' = \2 inches; 
ri = 18 inches. 

By making ^1 =0 in eqs. (12) and (13) of the preceding 
article there will result the following two formulas for the 
intensities p and h : 



Art. 41.] 



CYLINDER UNDER HIGH INTERNAL PRESSURE. 

^>2 ^'2 



P=-P 



■rr 



/2 



ri' 



219 



(8) 



.'2 



h = 



P—T, 



+ - 



(9) 



rr 



The two intensities p and k will be computed for six 
equidistant points, including those on the two cylindrical 
surfaces, by taking corresponding values of r. The results 
of these computations are given in the following tabulation 
and they are shown graphically in Fig. 2 , as will be explained 
further on. 







Pounds per Sq.In. 


r 


r't 




7 


rt 










P 


h 




I. 


-10.970 


+ 28,054 


I.I 


.8264 


- 7,543 


+ 25,093 


1.2 


.6944 


- 4.937 


+ 22,486 


1-3 


•5917 


- 2,908 


+ 20,459 


1-4 


.5102 


- 1,299 


+ 18,848 


1-5 


•4444 





+ 17,552 



Combined Cylinder under High Internal Pressure. 

The stresses induced by shrinkage in making the com- 
bined cylinder of two concentric shells have been explained 
and computed in the preceding sections; those stresses are 
permanent and they must be combined with stresses which 
may be produced usually temporarily by subjecting the 
combined cylinder to a high internal pressure such as that 
caused by the discharge of a gun. The internal pressure 



220 



HOLLOW CYLINDERS AND SPHERES. 



[Ch. IV. 



produced by a modern high explosive may reach 50,000 
or 60,000 lbs. per square inch, but as an illustration in this 
case the internal pressure will be taken as 40,000 lbs. per 
square inch. Hence the following data are required: 



^'=40,000 lbs. per square inch: 
Ti =iS inches. 



pi=o; / =6 inches; 



As this case is similar to that expressed by eqs. (8) and 
(9), those equations will yield the results shown in the 
following tabulation when the above data are substituted 
in them. 



r 


r'2 
r2 


Pounds per Sq.In. 


r' 


P 


h 


I I 




— 40,000 


+ 50,000 


I3 


5625 


-20,313 


+ 30,312 


I^ 


36 


— 11,196 


+ 21,200 


2 


25 


— 6,252 


+ 16,248 


2i 


1837 


- 3.268 


+ 13,268 


2f 


1406 


- 1,328 


+ 11,328 


3 


nil 





+ 10,000 



It will be observed that the intensities p and h have been 
computed at points 3 inches apart throughout the 12 -inch 
thickness of the combined cylinder wall. 

The results of computations shown in the three pre- 
ceding tabulations may now be shown graphically in Fig. 
2. That figure shows part of a normal section of the two 
cylinders, C being the center and CD being the internal 
radius of 6 inches. The separate walls each 6 inches thick 
are shown by the parts of concentric circles with radii 
6 inches, 12 inches, and 18 inches. The line ABD repre- 
sents the trace of a longitudinal diametral plane at right 
angles to which the intensities of the circumferential or 
hoop stresses showri in the preceding tabulations are laid off. 



Art. 41.] CYLINDER UNDER HIGH INTERNAL PRESSURE. 221 

Tensile stresses indicated by the plus sign are laid off to 
the left oi AD and compressive stresses to the right of BD 
as indicated by the minus sign. 

Referring to the tabulated results for the inner cylinder 
in conipression, DQ represents 29,250 and BP 18,283, both 
pounds per square inch. Intermediate ordinates of the 
curved line PQ are laid off by the same scale to represent 
the other intensities h given in the table. 

The ordinate AL represents by the same scale the 
intensity 17,552 lbs. per square inch and MB the intensity 
28,054 lbs. per square inch, both shown in the tabulation 
for the outer cylinder in tension. The other intensities 
laid off as ordinates give the curved line LM. 

The tensile intensities h for the combined cylinder under 
the internal pressure of 40,000 lbs. per square inch are 
shown by the ordinates to the curved line EF, FD repre- 
senting 50,000 lbs. per square inch and EA 10,000 lbs. per 
square inch. 

The resultant intensities at various points m the wall 
of the combined cylinder are found by taking the algebraic 
sum at each point of the three results shown. The result- 
ant hoop stress at D is found by laying off KF = DQ, 
the resultant intensity being 1)7^ = 50,000 — 29,250 = 20,750 
lbs. per square inch. Similarly BH =MB —BP = 16,248 — 
18,283 = —2035 lbs. per square inch, showing that the tensile 
stress developed by the high internal pressure was not 
quite enough to overcome the shrinkage compression. The 
intensities of hoop stress in the wall of the inner cylinder 
are therefore the intercepts of ordinates at right angles 
to BD between FM and KH. 

All stress in the outer cylinder is tension equal in 
intensity at any point to the sum of the ordinates between 
AB and ME added to those between AB and LS repre- 
sented by the ordinates drawn from AB to ON. Thus it 



222 



HOLLOW CYLINDERS AND SPHERES. 



[Ch. IV. 



is seen that the shaded parts of the diagram represent at 
each poirt the intensity of stress existing at that point. 




The highest tension exists in the outer cyHnder at B and is 
equal to 28,054 + 16,248=44,302 lbs. per square inch. At 



Art. 41] CYLINDER UNDER HIGH INTERNAL PRESSURE. 223 

the outer point A the tensile intensity of hoop stress is seen 
to be 27,552 lbs. The intensity of hoop stress at the 
interior surface of the cylinder has been found to be 20,750 
lbs. per square inch, materially less than at the outer sur- 
face, which is desirable, as the radial normal pressure at 
the inner point is 40,000 lbs. per square inch. 

The high tensile intensity 44,302 lbs. per square inch, 
found at the inner surface of the outer cylinder and the 
compression of about 2000 lbs. per square inch at the 
adjacent point on the inner cylinder show the desirability 
of a redesign for the assumed internal pressure with adjust- 
ment of the amount of shrinkage and with the wall com- 
posed perhaps of three cylinders instead of two. In this 
manner the undesired extremes of stress in the vicinity of 
the middle of the wall can be avoided. The results, how- 
ever, exhibit completely the procedures to be followed 
where it is desired to make a combined cylinder with a 
number of concentric shells with vshrinkage so employed as 
to produce a more nearly uniform, though not continuous, 
stress condition than can be attained in a single wall with- 
out shrinkage. In a single wall of 12 -inch thickness in 
this case the hoop tension would have varied from 50,000 
lbs. per square inch at the interior surface to only 10,000 
lbs. per square inch at the exterior surface. 

The radial compressive intensities p have not been 
plotted in Fig. 2, as the resultant intensity in every case is 
found by adding the intensities due to each condition as 
given in the tabulations. At the interior surface the maxi- 
mum intensity of pressure is 40,000 lbs. per square inch. 
At 3 inches from the interior surface the maximum inten- 
sity will be about 20,000 lbs. per square inch and at the 
common surface of the two shells that intensity will be 
about 17,000 lbs. per square inch, thus decreasing outward 
until the value o is found at the outer surface. 



224 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

Art. 42. — Thick Hollow Spheres. 

When the thickness of wall of a hollow sphere is so 
great that the stresses may not be considered uniformly 
distributed over a diametral section of the shell the approxi- 
mate formulae of Art. 39 cannot be used; it becomes neces- 
sary to make an investigation similar to that required for 
thick hollow cylinders. 

It will be supposed that the interior of the spherical 
shell is subjected to an intensity of pressure p' greater than 
the exterior normal pressure pi as shown in Fig. i. As 
the intensity of the interior pressure, produced possibly 
by a fluid, is greater than that of the exterior pressure the 
material of the shell will be subjected to an internal stress 
of tension as well as the radial compression, but the formulae 
as demonstrated will be equally applicable to the case of 
the exterior pressure being greater than the interior with- 
out any modification whatever. In the latter case, however, 
the internal stress acting around a great circle will be com- 
pression instead of tension. The formulas will be so written 
that a tensile stress is positive and a compressive stress 
negative. 

If a diametral section of the spherical shell be taken as 
in Fig. I, it is clear that for a given radius r there will be 
a uniform intensity of tension normal to that section and 
no other stress, i.e., this tension at every point will be in 
the direction of the circumference of a great circle. Fur- 
thermore, since that observation is true of all possible 
diametral sections of the shell it is equally obvious that at 
any point in the shell there will be two circumferential 
or hoop stresses at right angles to each other and a third 
radial stress of compression with no other stress on its 
surface of action, the three stresses being principal stresses 
at the assumed point. The three principal planes on which 



Art. 42. 



THICK HOLLOW SPHERES. 



225 



these principal stresses act are two of them diametral and 
at right angles to each other, while the third is tangent 
to the spherical surface with radius r, and it is at right 
angles to the other two planes. The state of stress in the 
interior of the shell is also obvious from the fact that the 
interior and exterior fluid or normal pressures are each the 
same^ in intensity at all points making the resulting con- 




FlG. I, 



dition of stress completely symmetrical. As every diam- 
etral plane section of the shell is a principal plane of stress 
there will be no shear on any such plane and for the same 
reason there will be no shear on any of the concentric 
spherical surfaces within the limits of the shell. 

Remembering that the interior radius of the shell is / 
and the exterior radius ri and that the tendency to tear the 
shell apart in any diametral annular section is due to the 
excess of the interior pressure over the exterior the follow- 
ing equation may be at once written, if h represents the 



2 26 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

intensity of the internal tensile stress developed at any 
point in the annular section: 

Tr{p'r'^-pir.i^)= ] h.2Trr.dr. . . . (i) 

If in eq. (i), r' and ri be considered variable and of 
so nearly the same value that they differ from each other 
only by dr, the quantity p'r"^ —piVi^ becomes equal to d{pr'^). 
Hence, eq. (i) for that supposition may take the following 
form : 

d(pr^) =2hrdr = 2prdr-\-r^dp (2) 

This is a differential equation between h and p. 
Another equation of condition is required to determine 
those two variable quantities. Such an equation may be 
written by so expressing the relation between the internal 
distortions or strains accompanying the stresses h and p 
as to make the diametral sections of the shell plane what- 
ever may be the intensities of the internal stresses h and p. 
The consideration of such relations between the strains 
produced would be precisely the same as given in Art. 8 
of Appendix I, and hence it is repeated here. If it be 
remembered that the intensities of the two circumferential 
stresses at any interior point of the shell are equal to each 
other and indicated by h, as in eqs. (i) and (2), while p 
represents the intensity of the internal radial stress at the 
same point, the relation between the internal strains or 
distortions necessary to make all diametral sections of the 
shell plane for all intensities of stress is equivalent to the 
condition that the sum of the three principal stresses must 
be constant at all points as expressed by eq. (3), a being 
constant : 

p + 2h=a (3) 



Art. 42.] THICK HOLLOW SPHERES. 227 

From eq. (3) : 

p=a — 2h and dp = 2dh. .... (4) 

Substituting from eq. (4) in the second and third 
members of eq. (2) : 

2 hrdr = 2 ardr — ^hrdr — 2 r^dh . 

By arranging terms the preceding equation takes in- 
tegrable form as given by eq. (5) : 

^hrdr-\-r^dh=ardr (5) 

If 6 is a constant of integration, eq. (5) may be integrated 
so as to take the form of eq. (6) : 



r^h=^-ar^-\-b', h=--\ — - (6) 

3 3 r^ 



By using the first of eqs. (4) and eq. (6), eq. (7) at once 
follows : 

^ a 2h . s 

'-s-y^ ^'^ 

At the inner and outer surface of the spherical shell 
p=p' and p=pi, respectively. Eq. (7) will then give: 

p'= ^andi7i=--— . .... (8) 

3 ^ 3 ^r 

Hence : 

I I \ ./^ — fi^ 



The preceding equation will at once give the following 
value of b, which in turn substituted in the second of eqs. 

8 will give the value of -, following that of b: 

3 

bJl^jy^; and«=^:!:;i:i^. . ; (9) 



228 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 



a 



These values of b and - substituted in eqs. (6) and (7) 

will give the following values of radial intensity p and cir- 
cumferential intensity h at any point in the shell distant 
r from the centre. In writing these final expressions it is 
to be remembered that the constants a and b may be either 
positive or negative and their signs are changed so as to 
make all positive stress tension and all negative stress com- 
pression, as was done in the case of thick cylinders. 



piTi^ -p'r'^ {pi -p')r'^ri^ i 



(10) 



These equations can be put in more convenient shape 
for computation by dividing all terms in the second mem- 
bers by ri^, which will give eqs. (loa) and (iia) : 



(loa) 



y'6 


{px-p'V^i 


P ,'3 


r'3 ^ 

{pi-p')r'^i 


'^- /3 


' /3 .^' 



(iia) 



It is necessary to determine a thickness / of shell which 
will resist a given intensity of internal pressure. Eq. (11) 
shows that the circumferential tension h will be greatest 
when r=r' in eq. (11). Making this substitution 

2h{/^-ri^)=Spiri^-p\2r'^+ri^). 



Art. 42.] THICK HOLLOW SPHERES. 229 

Dividing by /^ and solving: 

fi^^ 2(h-\-p') 
r'^~ 2h-p'+sPi' 

Hence there may be at once written: 



, ^ ,3 2{k+p') , ( . 

ri-r =t=r \ , , , r. . . . (12) 

yl2h-p'+spi 

This value of t will give the thickness of material re- 
quired so . that the maximum intensity of circumferential 
tensile stress shall not exceed a prescribed value h at the 
interior surface of the sphere when the interior pressure is 
p' and the exterior pressure pi, the latter being smaller 
than the former. 

A similar treatment may be given to eq. (11) after 
making r=ri in order to determine a thickness t such that 
the circumferential compressive stress shall not exceed a 
given prescribed value when the exterior pressure pi exceeds 
the interior pressure p'. 

In eq. (12) if ^' = 2/^+3^1, the value of t becomes 
infinitely great, showing that if the interior pressure reaches 
or exceeds the value indicated no thickness of shell what- 
ever will prevent the circumferential or hoop tension ex- 
ceeding the prescribed limit h. 

If either internal or external pressure become zero, 
while the other has any assigned value, it is only necessary 
to make either p' or pi equal zero in all the preceding 
equations. Furthermore, it is a matter of indifference 
whether p' or pi is numerically greater in the application 
of any of the preceding equations except eq. (12). 



230 HOLLOW CYLINDERS AND SPHERES. [Ch. IV. 

Radial Displacement at any Point in the Spherical Shell 

The general analysis of Art. 8 of Appendix I, gives an 
expression for the radial strain or displacement of the 
material at any point within the spherical shell. It has 
already been seen that no other displacement occurs, as 
all diametral sections of the shell remain plane for , any 
degree of stress whatever. If this radial displacement or 
strain at any point be indicated by p, the analysis indi- 
cated show^s that the value of this displacement will be 
given by eq. (13): 

^ 4<^l3(i+r) r;^_^ /^_ r^j' ' ^^^. 

t 

Knowing the internal and external pressures to which 
the shell is subjected eq. (13) will give the value of the 
radial displacement of any indefinitely small piece of 
material at the distance r from the center. If r = ri the 
corresponding value of p given by eq. (13) will indicate 
the increase or decrease, as the case may be, of the external 
radius ri; and if r=r' the increase or decrease in length 
of the interior radius r' will result. In eq. (13) G is ob- 
viously the modulus of elasticity of the material for shear, 
while r is the ratio of lateral over direct strains. 



CHAPTER V. 

RESILIENCE. 

Art. 43. — General Considerations. 

The term resilience is applied to the quantity of work 
required to be expended in order to produce a given state 
of strain in a body. If a piece of material is subjected 
to tension, that state of strain will be simply the stretching 
of the piece or the amount of compression, if the piece is 
subjected to compressive stress. In precisely the same 
manner the resilience of a bent beam is the amount of work 
performed upon it by its load in producing deflection. 
There may also be the resilience of shearing or of torsion. 

In the ordinary use of the expression, resilience refers 
to the amount of work expended within the elastic limit, 
whether of torsion, compression, or tension, but it m.ay 
properly be extended in its meaning to include the total 
amount of work required to rupture the material under 
any one of the preceding conditions of stress. Elastic 
resilience may easily be computed by means of exact 
formulas, but if the total work required to cause rupture 
in any case is desired, a graphical record of the total strains 
produced between the elastic limit and failure must be 
obtained by actual tests. In these articles the formulae 
for elastic resilience only will be given; in other subse- 
quent articles the method of computing the total resilience 

231 



232 RESILIENCE. [Ch. V. 

of failure will be illustrated by computations from actual 
strain records. 



Art. 44. — The Elastic Resilience of Tension and Compression 

and of Flexure. 

Let it be supposed that a piece of material whose length 
is L and the area of whose cross-section is A is either 
stretched or compressed by the weight or load W applied 
so as to increase gradually from zero to its full value. If 
E is the coefficient of elasticity, the elastic change of length 

will be -^-^. The average force acting will be Jl^, hence 

the work performed in producing the strain will be 

Resilience = — r^ (i) 

W 
If -J-, the intensity of stress in the metal, be represented 

by t, eq. (i) may be written 

Resilience =^Af^ (2) 

Again, eq. (2) may take the following form: 

Resilience = \AE^JL = ^AEPL. . . . (3) 

In eq. (3) the quantity /' = c^ is obviously the square 

of the strain (stretch or compression) per unit of length. 
If a bar of material i inch in length and i square inch 



Art. 44] RESILIENCE OF BENDING OR FLEXURE. 233 

in cross-section be considered, .4 = i and L = i must be 
inserted in the preceding equations, and there will result 

Unit resilience = ^-p=iEP (4) 

t^ 

The quantity -^ =EP is called the "Modulus of Resil- 
ience." The expression is ordinarily employed when t 
is the greatest intensity of stress allowed in the bar. 

The preceding equations are applicable whether the 
bar or piece of material is in tension or compression, the 
coefficient of elasticity E being used for either stress, while 
t represents the intensity of either tension or compression, 
as the case may be. 

Inasmuch as the values of t and E are usually taken 
in reference to the square inch as the unit of area, it is 
generally convenient to take L in inches, although any 
other unit of length may be taken when multiplied by the 
proper numerical coefficient. 



I he Resilience of Bending or Flexure. 

It has already been shown, in considering the common 
theory of flexure, as applied to the flexure or bending of 
beams, that the intensities of the stresses of tension and 
compression vary from point to point throughout the 
entire beam. In determining the elastic resilience of 
flexure, therefore, it is necessary to flnd the work per- 
formed in producing the varying strains corresponding 
to the stresses in the interior of the beam. The resilience 
due to the direct stresses of tension and compression will 
first be considered and then that due to the shearing 
stresses. 



234 RESILIENCE, [Ch. V. 

In order to obtain the expression for the work per- 
formed by the direct stresses of tension and compression 
in a beam bent by loads acting at right angles to its axis, 
a differential of the length, dL, is to be considered at any 
normal section in which the bending moment is AI, the 
total length of span or beam being L. Let / be the mom.ent 
of inertia of the normal section, ^4, about its neutral axis, 
and let k be the intensity of stress (usually the stress per 
square inch) at any point distant d from the axis about 
which I is taken. The elastic change produced in the 
indefinitely short length dL when the intensity k exists 

. k 

is pdL. If dA is an indefinitely small portion of the normal 

section, the average force or stress, either of tension or 
compression, acting through the small* elastic change of 
length just given, can be written by the aid of a familiar 
equation of flexure as 

1 7 r. Md 
-k.dA=—f 

2 21 



k.dA=^f.dA . (5) 



Hence the work performed in any normal section of the 
member, for which M remains tmchanged, will be, since 

Jk.dA.d=M, 



I 



M M' 
^^^kd.dA.dL=-^^dL (6) 



The work performed throughout the entire piece will then 
be 

wi"- I" 



/ 



The integration indicated in eq. (7) is readily made 
in all ordinary cases by substituting the value of the bend- 



Art. 44.J RESILIENCE OF BENDING OR FLEXURE. 235 

ing moment M in terms of the variable horizontal ordinate 
or abscissa x and the load, it being remembered that dL 
is precisely the same as dx. If, for example, the beam 
is non-continuous, simply supported at each end and 
carries uniformly distributed load p per unit of length 

P 
throughout the whole span, AI =-{Lx — x^). By the in- 
sertion of this value of M in eq. (7), there will result 

I r^ pV p^D W^L^ 

Resilience = —j=rf I ~ — (L — xydx= ^ -^-. = =— , (8) 

2EIJ0 4 ^ ^ 240EI 24oEr ^ ^ 

W representing pL, the entire load on the beam. 

This equation gives the value of the total work per- 
formed by the direct stresses of tension and compression 
in the interior of a simple beam uniformly loaded and 
supported at each end, under the assumption that the 
moment of inertia / of the cross-section is constant through- 
out the entire span. 

If a single load W rests at the centre of the span, the 

W 
reaction at each end being — , the value of the bending 

W 
moment at any point will be — x. By inserting this value 

of ikf in eq. (7), there will result 

Kesihence=—j=^.2 / — x^ax=-—r=^. . . (9) 
2EI Jo 4 g6EI 

Similarly equations of the elastic resilience of the direct 
stress of tension and compression in beams loaded in any 
manner whatever may easily be written. In some cases 
like the last the deflection at the point of application of a 
single load may easily be determined. Let that deflec- 



236 RESILIENCE. [Ch. V. 

tion be represented by w\ when a single load W rests at 
the centre of the span the work performed by this load in 
producing the deflection is \Ww. Hence that amoiint of 
work must be equal to the resilience given by eq. (9), and 

WU 
w= j-,j (10) 



The Resilience Due to the Vertical or Transverse Shearing 
Stresses in a Bent Beam. 

The work performed by the vertical shearing stresses 
in a bent beam of any shape of cross-section may readily 
be found. Let 5 be the total transverse shear in a normal 
section, I being the moment of inertia of the latter about 
its neutral axis, h the width or breadth (constant or variable) 
of the section, ^ the unit shearing strain defined in Art. 2, 
d and d^ the distances of the extreme fibres from the neu- 
tral axis, and G the coefficient of elasticity for shearing. 
By eq. (6) of Art. 15 the intensity 5 of the shear at any 
point in the section at the distance z from the neutral axis 
will be 

S^j^{d^-Z^) (II) 

Again, by eq. (3) of Art. 2, 

*-i-4ri^^'-''^ • (-) 

The amount of shearing stress on the indefinitely small 
portion of the section h.dz will be sh.dz, and its path in 
performing the w^ork will be (t>dx, x being the horizontal 
ordinate of the section of the beam from any convenient 
origin, as the end of the centre of the span, i.e., in this case 



Art. 44-] RESILIENCE DUE TO SHEARING STRESSES. 237 

the end of the span. The differential work performed in 
the section will be, by the aid of eqs. (11) and (12), 

iisbdz).{^dx)--^^j^^{d'-zydzdx. . . (13) 

In this eqtiation it is easy to express the breadth b of 
the section in terms of z, whatever may be its shape, by 
the aid of the equation of the perimeter of the section. 
In all the ordinary and important cases of engineering 
practice involving this resilience of shearing the shape of 
the section is rectangular for which b is constant, and it 
will be so regarded in the following equations. Remem- 
bering that X and z are independent variables, and that 
the first integration will be made in reference to z, that 
integration will give 

^fsH.fy^^.Yd.^-^^^fsHx. (Z4) 

As the section is taken to be rectangular in outline, with 

h bh^ 

the breadth b and depth h, d=d^=~ and /=- — . Eq. (14) 

will then become 

Resilience =-j-r^ / S^dx (15) 

The total transverse shear 5 will have varying values 
depending upon the amount of loading on the beam and 
its distribution, i.e., in general it will vary with x, and 
when not constant it must be expressed in terms of that 
variable before the remaining integration can be made. 

If a single weight W rests on the beam at the distance 
of V from one end where the reaction is R', and at the 



23^ RESILIENCE. [Ch. VI. 

distance /^ from the other end where the reaction is R^, 
the shear S will be constant for each of the segments into 
which the point of loading divides the span ; in one of 
those segments S = R\ and in the other S = R^. The com- 
plete integration of eq. (15) will be, therefore, 

Resilience = -^(R''r+R^\). 

If there be substituted in the parentheses of the second 
member of the preceding equation the values R' =W-r 

V 
and i?i = Wj, there will result 

3 ir 

Resilience = 7 , ^ -V W^ (16) 

SbhG I ^ ' 

If the weight W rest at the centre of the span l^=r = - 
and 

Resilience =—~tWH (17) 

Eq. (17) affords a simple method of finding the deflec- 
tion w^ of the point of loading due to the transverse shear. 
As the weight W is supposed to be gradually applied the 
expended work JT/Fw^ must be equal to the shearing re- 
silience given in eq. (17). Hence 

""'^i^il ^^8) 

When a non-continuous beam simply supported at each 
end carries a uniform load over the entire span, it has been 
shown in Art. 22, eq. (7), that the transverse shear at any 



Art. 44.]. DIRECT AND SHEARING STRESSES. 239 

section is equal to the load between the centre of span and 
that section. If, therefore, the origin of x be taken at 
th-e centre of span and if p represents the load per unit of 
length of the beam, S=px. By substituting this value of 
5 in eq. (15), and remembering that twice the integral 
must be taken for the whole beam, 

Resilience = —t^tt ■ 2 / p x^dx = ^ ^^ ^ = ^ — — — fi q ) 
SGhh Jo ^ 2oGhh 2oGhh' ^^^ 

The shearing resilience, therefore, in a non-continuous 
beam carrying a uniform load is only one third as much 
as that due to the same load concentrated at the centre of 
the span. 

If, as is usual, (7 is expressed in pounds per square inch 
the unit for /, 6, and h will be the linear inch. 

Other modes of loading than those takeji can be treated 
in precisely the same general manner. 

As the intensity of the longitudinal shear at any point 
of a beam is the same as that of the transverse shear, 
the total work of the longitudinal shear throughout 
the beam is the same as the work of the transverse 
shear. The total work of the shearing stresses in a bea.m 
is therefore composed of those two equal parts. 

The Total Resilience Due to Both Direct and Shearing 

Stresses. 

The general expression for the total resilience of a bent 
beam due to both shearing and direct stresses will be the 
sum of the second members of eqs. (7) and (13), expressed 
by the following equation:. 



Total resilience = I —FrjdLi- I I of27^(<^' 



z^) "^dzdx. 



240 RESILIENCE. [Ch. V. 

Or, by eqs. (7) and (15), since dL=dx, 

Total resilience = / ~cjd^-^7n~Y- S^dx. . (20) 

By the aid of eqs. (8) and (19) the total resilience for 
a simple non -continuous beam may be as follows: 
If the imif orm load pl = W, 

Total resilience = W^{j^^ + ^^. . (21) 

For the same beam carrying a single load W at the centre, 
by eqs. (9) and (17) 

Total resilience = W^(^^^ + -^^. . . (22) 

As has been explained, the last two equations are appli- 
cable to beams with rectangular sections only. 

In a similar manner the total deflection of a beam 
supported at each end and loaded with a single weight W 
at the centre of the span, due to bending and flexure, will 
be found by the sum of the two expressions given in eqs. 
(10) and (18): 



Art. 45. — Resilience of Torsion. 

The w^ork expended in producing elastic strains of 
torsion constitutes the resilience of torsion and is a special 
case of shearing resilience. The twisting moment which 
produces the angle of torsion a is given by eq. (i6) of 



Art. 44-] RESILIENCE OF TORSION, 241 

Art. 37 and is iTf=6'a/^. When the piece twisted has the 
length / the total angle of torsion is al and the differential 
amount of work performed by the moment M in producing 
the indefinitely small twist d{al) =l.da is Ml. da. Hence 

Resilience = I Mlda=GlIp / a . da = Gil p—^ . . (i) 

If P and e are the force and lever-arm of the twisting 
couple, eq. (18) of Art. 37 shows that 



^^^SZ 



Pe^ 
p 



Substituting this value of a^ in eq. (i), 



P'^eH 
Resilience = ^r (2) 

2Ul ^ ^ 



p 

4 



7ZT 

If the normal section of the piece is circular Ip = 

Hence, for a shaft with circular section, 

P^-eH 
Resilience = -7:^—^ (3) 

If the section of the shaft is a square, Ip = — , b being 
the side of the square. Hence, for a square section, 

Resilience = r-14 (4) 

In some cases shafts are subjected to combined torsion 
and bending. In such cases, if it is desired to compute 
the total elastic resilience it is only necessary to take the 



242 RESILIENCE. [Ch. V. 

sum of the two resiliences, each foiind as if existing in> 
dependently of the other. 

The resiHence of torsion beyond the elastic limit or 
between the elastic limit and the ultimate resistance must 
be determined, as in all cases of distortion beyond the 
elastic Hmit, from an actual strain record, as given, by 
the testing machine when the piece is strained up to any 
given degree of permanent stretch or to rupture. 

Art. 46. — Suddenly Applied Loads. 

A load is considered suddenly applied when its full 
amount acts instantly upon any piece of material loaded 
by it. In the preceding articles relating to resiHence the 
loads are treated as being gradually increased from zero 
to their full values. In such cases the amount of external 
loading at any instant is supposed to be equal only to the 
internal stress or stresses opposing it, so that the w^ork 
performed is equivalent to one half the total load multi- 
plied by the total resulting strain. When the loads are 
suddenly applied, on the other hand, the internal stresses 
produced are exactly equal to the external forces only 
when the strains corresponding to the latter are reached, 
and the work performed up to that point is just double 
the work expended when the loads are gradually applied. 
It follows from this last . consideration that the strains 
produced by the suddenly applied loads will be double 
those found under gradual application. Inasmuch as 
the elastic strains are proportional to the corresponding 
stresses, it further follows that the stresses produced by 
suddenly applied loads will be double in intensity those 
which are produced by the same loads gradually applied. 

The work expended by a suddenly applied load up to 
the point of strain corresponding to its amount being 



Art. 46.] PROBLEMS FOR CHAPTER V. 243 

double the work performed by the internal stresses, the 
total stress induced in the material at the limit of the final 
strain produced by such a load will be double the amount 
of the latter. The internal stresses in the piece will, there- 
fore, cause it to recover from its strained condition and 
vibrations will result, the treatment of which constitutes 
an important branch of the theory of elasticity in solid 
bodies. Some general features of that treatment will be 
given in Art. 12, App. I, but as they are seldom used in 
engineering practice they will not be considered here. 
It is only important at this point to note carefully 
the distinction between the effects of a given load grad- 
ually applied and suddenly applied, the strains and 
stresses in the latter condition being double those in 
the former. 

Again, it is also important to distinguish between 
loads suddenly applied, and shocks, as they are called in 
engineering practice. A shock is produced when the 
load falls freely before acting upon a piece of material 
sustaining it.- The cause of shock, therefore, is a suddenly 
applied load with the effect of a free fall of the latter super- 
imposed. These matters must be carefully taken into 
account and allowed for in such structures as bridges 
carrying rapidly moving trains, and those allowances are 
incorporated in the provisions of specifications covering 
bridge construction. 

Problems for Chapter V. 

Problem i. — A 6-inch by 1.7 5 -inch steel eye-bar 48 feet 
long is subjected to a stress of 117,500 pounds. If that 
load is gradually applied what is the work performed in 
the total length of the bar, if £ = 30,000,000 pounds? Also 
what is the unit resilience? 



244 RESILIENCE, [Ch. V. 

t = — — — = 11,190. L = 48X12 =576 inches. Eq. (2) 

of Art. 44 then gives 

r> -r 1 ^ A' io.5X(ii,i9o)'X576 

Resihence = work performed = ■ 

^ 2X30,000,000 

= 12,621 in. -lbs. 
Eq. (4) of Art. 44 gives 

Unit resihence = ' =2.00 in. -lbs. 

2X30,000,000 

Problem 2. — A cast-iron column 18 feet long having 
an area of cross-section of 40.8 sq. in. carries a load of 
245,000 pounds. If the coefficient of elasticity E is 14,- 
000,000 pounds, how much work is performed in com- 
pressing the column if the load is gradually applied. 

Problem 3. — A 30-pound lo-inch rolled steel I beam 
carries a uniform load of 1000 pounds per linear foot in 
addition to its own weight with a span of 16 feet. What 
will be the resilience or work performed in the material 
of the beam imder the gradual application of that total 
load of 1030 pounds per linear foot, the moment of inertia 
/ of the beam being 134.2 and £ = 30,000,000 poimds? 
Eq. (8) of Art. 44 is to be used, in which L is 192 inches. 
Incidentally, what will be the greatest intensity of stress, 
k, in the extreme fibres?. 

Ans, Resilience = 1987 in. -lbs ; k = 15,000 lbs. per square 
inch. 

Problem 4. — In Problem 3 if the thickness of the web 
of the lo-inch rolled beam is .5 inch, find the resilience oif 
the vertical or transverse shearing stresses in the beam, 
the coefficient of shearing elasticity, G, being taken at 
12,000,000 pounds. The remaining data are / = i92 inches; 
h = io inches; ^=0.5 inch, and 1/^ = 16,480 pounds, and 
they are to be used in eq. (19) of Art. 44, 



Art. 46.J PROBLEMS FOR CH/fPTER V, 245 

Problem 5. — A round bar of steel 2I inches in diameter 
is twisted by a force of 2100 pounds acting with a lever- 
arm of 17 inches. Two sections 25 ft. apart are turned 
0.185 inch in reference to each other, i.e., the total strain 
of torsion for a length of bar of 25 feet has that value. 
Find the total angle of torsion, the angle of torsion and the 
coefficient of elasticity, G, for shearing (i.e., for torsion). 
Ans. 01=0.00043; q:/ = 0.129; and 6' = 13,000,000 lbs. 

Problem 6. — The greatest permitted working intensity 
of torsive shearing is 8000 pounds per square inch. Design 
a steel shaft to carry a twisting moment produced by a 
force of 1900 pounds, acting with a lever-arm of 84 inches. 
If the coefficient of elasticity for shearing is 12,000,000 
potmds, what will be the angle of torsion? Also what will 
be the total angle of torsion and total strain of torsion for 
a length of shaft of 13 feet? 

Problem 7. — In Problems 5 and 6 find the work per- 
formed in twisting the two steel shafts, i.e., the resilience 
for 25 feet length in the one case and 13 feet in the other. 
Use equations of Art. 45. 

Problem 8. — In Problem 5 suppose the load suddenly 
applied, what will be the resulting resilience and greatest 
intensity of extreme fibre stress? 



CHAPTER VI. 
COMBINED STRESS CONDITIONS. 

Art. 47.— Combined Bending and Torsion. 

Probably the most important case of combined bend- 
ing or flexure and torsion is that of the ordinary crank- 
shaft represented in Fig. i. 

The centre of the thrust of a connecting-rod is at A, 
on the crank-pin journal against which the connecting-rod 
bears. The centre of the shaft-bearing is at B. If the 
thrust at A is represented by P, then the actual resultant 
moment about the centre of the bearing B will be PxAB. 
The problem is to determine the maximum stresses de- 
veloped by this resultant moment in the section of the 
shaft at B. Two methods may be employed in both of 
which the resultant moment of P multiplied by the lever 
arm AB is resolved into its two components, one of which 
is the ordinary bending moment represented by M =Px CB, 
and the other is the twisting moment M' =PxAC. The 
latter produces torsion in the journal at B and the former 
produces pure flexure or bending at the same section. 

Let CB be represented by / while e represents AC. 
The moment of pure bending at B will be 

■ . M=Pl. ....... (i) 

246 



Art. 47.] COMBINED BENDING AND TORSION. 247 

The twisting moment producing pure torsion will be 

M'^Pe (2) 

If d represents the distance of the most remote fibre 
in the" section B from the neutral axis of the latter, and if 




/ 



Fig. I 



k is the greatest intensity of bending stress at the dis- 
tance d from the neutral axis, while / is the moment of 
inertia of the normal section of the shaft at B about the 
same neutral axis, the following will be the value of k : 

Md_Pld 
I ~ I ' • • • • ° 



k = 



(3) 



Again, if T is the greatest intensity of torsional shear 
in the normal section of the shaft at B, at the greatest 
distance r, in the perimeter, from the centre of gravity or 
the centroid of the same section, the value of the maximum 
intensity T will be 

_ MV Per 



(4) 



In eq. (4) Ip is the polar moment of inertia of the nor- 
mal section at B. 



248 COMBINED STRESS CONDITIONS. [Ch. VI. 

First Method. 

In this method it is only necessary to consider the 
intensities k given by eq. (3) and T given by eq. (4), the 
greatest allowed working stresses of direct tension and of 
shearing respectively, k would have the value of the 
greatest tensile working stress of the material of the shaft 
for the reason that if tested to failure the shaft would 
yield first on the tension side. 

It being understood, therefore, that ^ and T represent 
the greatest allowed working intensities of stress, usually 
expressed in pounds per square inch, eq. (3) will give 



(s) 



/ 


M 
" k " 


PI 




d^ 


'' k ° • 




Under the same conditions < 


eq. (4) will give 




h_ 


M' 


Pe 




r 


" T^ 


^ T 


. 


For the circular section 






4 


and 




> • 


For a square section 








12 


and 


i^-^, . . . , 


. 



(6) 



(7) 



(8) 



b being the side of the square. In eq. (5) for a circular 
section d^r and for a square section d = — ;=. In eq. (6), 
r = r for the circular section, but for the square section 



Art. 47.] 



COMBINED BENDING AND TORSION. 



249 



r = -—=i. Making those substitutions in eqs. (5) and (6) for 

V 2 

the circular section, there will result, D being the diameter 
of the. shaft. 



For bending , . .r =— =^1 _^ = 1.08-^ 



D 
2 



PI 

k 



D shPe ^,sfPe 
For torsion . „ , . r =— ^\~^ =.86a(-y 



(9) 



In the practical use of eqs. (9) that one of the two 
values of r should be taken which is the greatest. This 
will insure that both the direct stress of tension and the 
shearing stress shall not exceed the prescribed values of 
k and r. 

The substitution of the values of / and Ip for the square 
section in eqs. (5) and (6) will give, remembering that d 
and r are each one half the diagonal of the square, 



3 6\/2Pl 



For bending . . . 6 = \ 



For torsion . . . .b = i.62\(^ 



= 2 



.04^1 



PI 
k 



(10) 



In eq. (10), also, the greatest value of h given by the 
application of the two formulae is to be taken, so that, as 
in the case of the circular section, neither of the two in- 
tensities k and T shall exceed the values prescribed for 
them. 

This method involves only the consideration of the 
simple formulae of the common ^theories of flexure and 
torsion. 



250 COMBINED STRESS CONDITIONS. [Ch. VI. 

Second Method. 

The second method of treatment of this case of the 
crank-shaft consists in determining the greatest intensity 
of the direct stress of tension in the section B of the shaft 
at the jom*nal-bearing. This resultant maximum intensity 
is produced by the combination of the same component 
moments, M =Pl and AF =Pe, as in the preceding method. 
With the sections of shafting ahvays employed the maxi- 
mum intensity of bending stress k and the maximum 
intensity of torsional shear T exist at the same point and 
on the same plane, i.e., the plane of normal section of the 
shaft. The existence of the shear T on the normal section 
at the distance r from its centre of gravity carries with it 
the same intensity of shear at the sam.e point on a longi- 
tudinal plane passing through the axis of the shafting. 
At the point considered, therefore, on two indefinitely 
small planes at right angles to each other, one normal to 
the axis of the shaft and the other parallel to it, there exist 
the direct intensity of tension k on the first, and the 
intensity of shear T on the second. The problem is to 
determine at the same point the greatest intensity of 
the direct stress of tension on any plane whatever, and 
the angle /? between the direction of that stress and the 
axis of the shaft. Reference may best be made to the 
general fonnulae of internal stresses in a solid body for its 
solution, and those are eqs. (8) and (9) of Art. 8. Those 
equations are adapted to this case by making px=k, 
pxu = T, tan a=tan /S, and p = t, the latter quantity being 
the greatest intensity of tension desired. These substi- 
tutions give the following two equations : 



/=J^ + 72 + ^ -(ii) 

\ A 2 



Art. 47.] COMBINED BENDING AND TORSION. 251 

27 
k 



tan 2/3 ^-— (12) 



Eq. (11) gives the greatest intensity of direct tension 
,in the shaft in terms of known stresses. 

By eq. (12) the position of the plane or section of the 
shaft on which the maximum intensity t exists may at 
once be found. Inasmuch as ^ is the angle between the 
direction of the stress t and the axis of the shaft, the angle 
between the plane on which t acts and the axis of the shaft 
will be 90°+^. 

Under this method of treatment it would be necessary 
to design the shaft so that t should not exceed the greatest 
prescribed tensile working stress for the material em- 
ployed. 

The greatest intensity of compressive stress in the shaft 
would be found by giving the negative sign to the radical 
in the second member of eq. (11). 

The preceding formulae have been established in a 
manner to make them applicable to any form of shaft 
section or any values of k and T. It is only necessary to 
insert in those formula any intensities of those stresses 
that may exist. If, for example, it were considered desir- 

P 

able to add the shear — -o due to the thrust P to the tor- 

Tzr 

P 

sional shear it would only be necessarv to take T -\- — -, 

for T wherever the latter quantity occurs. 

If a shaft is circular in section, as is almost universally 
the case, so that Ip ==2/, and if the shearing effect of P in 
the section at B, Fig. i, be omitted, useful and extremely 
simple relations may be deduced. In that case D = 2r, 
being the diameter of the shaft, and j the angle ABC 



2S2 COMBINED STRESS CONDITIONS. [Ch. VI. 

cf Fig. 1, M as before being the resultant moment, or 
M=PXAB: 



, rM cos y ■ , ^ rM sin / , , 

k = J — ^ and T = -j-^. . . (13) 



By the substitution of these values in eq. (11), 



^ = ^(i4-cos;)=--^(i+cos;). . . (14) 
Hence 



D = i.72^J-j{i+cosj) C15) 



Eq. (14) gives, by the aid of the first of eqs. (13) ^ 



^ = -^(i+cos y) =-(sec y + i). . . . (16) 
21 2 



The second of eqs. (13) gives, after substituting the 

value of — ;: = 



2/ irD^' 



^ ^.iM sin / , V 



The substitution of the values of T and k from eqs. (13^ 
in eq. (12) gives 



Art. 47] 



COMBINED BENDING AND TORSION. 



tan 2i 



T 



= tan j ; 



i/. 



253 

. (18) 



This last set of results relating to circular shafts will, 
in all ordinary cases, supply everything required for the 
operations of design or of investigations regarding con- 
ditions of stress in existing shafts. 

Eqs. (13), first of (14), (16), and (18) apply as they 
stand to square shafts. 

The first method involves simpler considerations than 
the second, not only analytically, but also in respect to 




Fig. 2. 

empirical quantities required to be used. The test pieces 
from which the ultimate resistance of the material is de- 
termined are always taken parallel to the axis of the shaft, 
but the greatest intensity of stress t found in the second 
method has a direction inclined to that axis by the angle /?. 
In general, therefore, it will probably be found more 
practicable to use the first method rather than the 
second. 

In the case of the double crank-shaft shown in Fig. 2, 
it is only necessary to treat each half precisely as if it were 
the single crank-arm in Fig. i. 



2 54 COMBINED STRESS CONDITIONS. [Ch. VI. 



Art. 48. — Combined Bending and Direct Stress. 

There are a considerable number of practical problems 
of combined flexure and direct stress of sufficient impor- 
tance to merit careful examination, and among them is the 
flexure of long columns treated in Art. 24. In this place 
the particular cases to be considered are those in which the 
bending is produced by a uniform load at right angles to 
the axis of the member, or by eccentricity of longitudinal 
loading, the direct stress (or external force) being applied 
in a direction parallel to the same axis. Lower chord 
eye-bars and. other horizontal or inclined chord members 
of 'pin bridges belong to this class. 

Let ilfj represent the bending moment in the m.ember 
at that section where the deflection is greatest, produced 
by loading at right angles to the m.ember's axis or by 
eccentricity in the application of the longitudinal loading; 
let w^ represent the greatest deflection resulting from the 
total bending moment and direct stress ; also, let P be the 
total direct stress acting upon the member whose length 
is /, while k represents the greatest intensity of stress due 
to bending alone and at the distance d of the most remote 
flbre from the neutral axis of the section at which the 
deflection w' is found. Finally, let A be the area of cross- 
section of the member which, together with the moment of 
inertia I, is supposed to be constant throughout the entire 

P 

length; and let <?=^, the intensity of uniform stress in 

the member due to the direct stress or force P. 

The rcsaltant mcximum bending moment in the 
member will then be 

M=MidzPw' (i) 



Art. 49.] EYE-BAR SUBJECTED TO BENDING. 2$$ 

If P is tension it will tend to pull the member straight, 
thus producing a moment opposite to M^. In the second 
member of eq. (i), therefore, the negative sign is to be used 
for a member in tension and the positive sign for a member 
in compression. 

The greatest resultant intensity of stress, t, in the 
member will then take the value 



P Md _^l ^ Md 

3+ / ~A 



^=^+^=4(P+^) (2) 



The quantity r is the radius of gyration, so that 

I=Ar\ 

When the intensity t is prescribed, the required area 
of section A is 

--K--^1 <3) 

These equations are perfectly general and may be 
applied to all cases of combined bending and direct stress. 

Art. 49. — The Eye-bar Subjected to Bending by Its Own Weight 
or Other Vertical Loading. 

Let Fig. I represent a low^er chord eye-bar of a pin- 
connected bridge with the length / and carrying the total 
tension P. The depth of the bar is h and the thickness h, 
so that the area of the normal section is hk. The bar acts 
as a beam carrying its own weight as a uniform load over 
the span /. That load deflects the bar as a beam while the 
direct stress of tension (P) decreases that deflection by 
tending to pull the bar straight. The problem is to deter- 



256 COMBINED STRESS CONDITIONS. [Ch. VI. 

mine the greatest stress in the bar and incidentally its centre 
deflection. 

There are several methods of procedure. The first and 
simplest method is approximate in its results, although 
sufficiently close for some purposes. It consists in treatmg 





Fig. I. 

the bending and direct stresses as existing independently, 
so that results are obtained by simply adding the bending 
to the direct intensities. This method will be treated 
first. 

The more exact method consists in recognizing the bend- 
ing moment as the resultant of those due to the transverse 
load acting on the bar as a simply supported beam, and to 
the direct stress P acting with the greatest deflection as 
its lever-arm. 

Approximate Method. 

Although reference will be made to Fig. i, the formulas 
as written will be equally applicable to compression mem- 
bers in which P would be the total force of compression. 

If the total weight of the bar or compression member 
is W, and if / is the moment of inertia of its cross-section 
about the neutral axis, while k is the greatest intensity of 
bending stress at the distance d from the same axis, the 
theory of flexure gives 

,, Wl kl ^ Wld 



Art. 49.] EYE-BAR SUBJECTED TO BENDING. 257 

If the area of cross-section is represented by A, while 
the radius of gyration is r, I=Ar'^. Again, the quantity 
I^d is called the "section modulus," and tabulated 
values of it for rolled sections may be found in hand-books. 
Let m be that modulus, then eq. (i) may take the form 

Wld Wl 
^~SAr'~Sm ^^^ 

The intensity of direct tension is 

^=z- (3) 

Obviously k will be tension on the lower side of the bar 
or other member and compression on the upper side. The 
greatest intensity of stress in the piece will be the sum of 
q and k. Eqs. (2) and (3) will, therefore, give the value 
of that greatest intensity, /, of stress as follows: 

, I /^ Wld\ 
t=q + k = ^[^P + ^).. . ... (4) 

When the greatest value of t is prescribed, the required 
area of section. A, can be at once written from eq. (4) 

i/^ Wld\ , , 

+ ^^2) (5) 



In the case of an eye-bar with the cross-section bh, 



d=— and -7 =t- Hence 
2 r^ h 



I 



3M\ 



bhy ^ 4h/ 
and 



t^TTAP + '-ir) (6) 



bh 



=K-?^) <'> 



258 COMBINED STRESS CONDITIONS. [Ch. VI. 

If the bar carries any other uniform load than its own, 
it is only necessary to make W represent the total uniform 
load, including the weight of the bar itself. 

Finally the direct force P may act with the eccentricity e. 
In this case the moment Pe produces uniform bending 
throughout the length of the bar, and it is only needful to 

write (-^±Pej for -^ m the preceding formula, the 



double sign showing that Pe may act either with or against 
the moment of the uniform load. 

The formulas of this article are not sufficiently exact 
for the usual cases of engineering practice. 

Art. 50. — The Approximate Method Ordinarily Employed. 

The method commonly employed in practical work for 
the treatment of compound bending and direct stress is 
a much closer approximation than the preceding method, 
although not exact. Its chief f capture is the introduction 
of the bending moment produced by the direct or longi- 
tudinal force multiplied by the actual maximum deflection. 
In the same manner the moment due to the eccentricity 
of the line of action of that force is introduced wherever 
necessary. 

Eq. (6a) of Art. 27 gives the following expression for 
the deflection w' due to pure bending and in terms of the 
greatest intensity of bending stress k, a being a constant 
depending, among other things, upon the distribution of 
loading : 

^=^E5 ^'^ 

If the deflection as given in eq. (i) be placed equal to 
each of the two parts of the deflection given in eq. (21) 



Art. 50.] APPROXIMATE METHOD ORDINARILY EMPLOYED. 259 

of Art. 28, it will be found for a beam simply supported 
at each end and loaded uniformly, that a=i\, and for 
the same beam loaded by a single weight only at the centre 
of the span, a=^g. The cases which occur in practice 
conform nearly to that of a load uniformly distributed 
over the length /. Hence for such a beam there is ordi- 
narily tak<sn 

^ =7^£d (^) 

The moment produced by the direct force or stress P 
acting with the lever arm w' will have the opposite sign 
to that of ilfj (the moment due to transverse loading or 
to eccentricity), if the member is in tension, but if the 
member is in compression those two moments will have 
the same sign. The resultant equation of moments may, 
therefore, be written 

M~=M^±P^w' (3) 

As stated, the plus sign is to be used for a compression 
member and the negative sign for a tension member. 

If the value of zu^ given by eq. (2), be substituted in 
eq. (3), the following value of k will result: 

, MJ 

(4) 



lot. 



In eq. (4) the plus sign is to be used for tension mem- 
bers and the minus sign for compression members. This 
equation is general and adapted to all forms of cross- 
section under the conditions virtually assumed. Although 
not explicitly stated, it is essentially assumed that the ends 



26o COMBINED STRESS CONDITIONS. [Ch. VI. 

of the member remain absolutely fixed in distance apart. 
This is frequently not the case, especially in the lower 
chord eye-bar of a pin -connected bridge subjected to direct 
tension and to bending due to its own weight, the bar 
usually being horizontal. 

If the ends of the beam or member, uniformly loaded, 
are fixed, a =3^, when k is the greatest intensity of bending 
stress at the mid-point of the member, or ^j if k is the 
intensity of the bending stress at the fixed ends. One of 
those values (usually -Jg) is to be substituted therefore 
for yV in the formulas which follow when the fixed-end 
condition exists. 

The resultant maximum intensity of stress t in the 
member will obviously be 

t=k + q (5) 

in which equation q is the uniform intensity P ^A. 

Eq. (4) will be immediately applicable to any particu- 
lar case by substituting in it the values of I and M^ for 
that special case. 

If the case of the lower chord eye-bar mentioned in a 
preceding paragraph be considered, the total weight of the 
bar being W, while b and h represent its thickness and 

depth respectively, 7= — • and M^=~q-. These values 

substituted in eq. (5) will give the desired value of the 
resultant intensity, as follows: 

(6) 

^^hh'^J~^ Wr ^ ^ 

3 SEh' 



,2 



Eq. (6) gives the value of trie maximum intensity of 
tension in the extreme lower fibres of the eye-bar when 



Art. 50.] APPROXIMATE METHOD ORDINARILY EMPLOYED. 261 

subjected to the total direct tension P and to the bending 

due to its own weight. 

The greatest intensity of bending stress in the bar is 

evidently the second term of the second member of eq. (6), 

and it has the following value if the weight of the bar per 

W 
unit of length is ~r =g, or if the weight of a cubic unit of 

the metal is i: 



8 h lihE 



6 P^bh 6^P'^'^ 



(7) 



It is frequently important to observe what depth of 

bar with a constant area of cross-section, subjected to a 

prescribed working stress, will give the maximtim bending 

stress due to its own weight when the length is fixed. 

That depth can readily be determined by taking the first 

derivative of k, as given by eq. (7), with h as the variable. 

dk 
By performing that operation and placing jt. = o, there 

will at once result 

^=vb^^ («) 

The value of h resulting from an application of eq. (8) 
gives the depth of bar which, with a given value of /, will 
under the conditions of the case yield the greatest in- 
tensity of bending stress k; it indicates, therefore, a limit 
of depth to be avoided as far as practicable. 

Steel is the usual structural material for eye-bars for 
which E may be taken at 29,000,000. For this value of 
E, h will become, by eq. (8), 

h = \/q7 

4900 ^ 



262 



COMBINED STRESS CONDITIONS. 



[Ch. VI. 



p 

In this equation q^-^ is the intensity of uniform stress 

in the bar, or the ' ' working stress. 

By placing the value of h, as given by eq. (8), in the 
value of k, eq. (7), there will result the maximum possible 
bending stress in a bar of given length / and given area of 
cross-section A : 



■VE 



^ I 



Vq 



(9) 



If £ = 29,000,000 and ^"-=.286 lb. per cubic inch for 
steel, eq. (9) will take the value, for the corresponding 
values of h in the equation preceding eq. (9), 



5£o/ 



(10; 



The following table shows at a glance the greatest 
possible fibre stresses in eye -bars of different lengths and 
depths when the working tensile stresses in pounds per 
square inch are those given in the extreme left-hand column 
of the table : 







Length of Eve-bars in F 


eet. 








Workinj? 
















Tensile 


















Stresses 


15 


20 


25 


30 




- 







m 
Pounds 










































per 




ui • 




'i. 




i/i 




m 




■/. 




t/i 


Square 
Inch. 

































° 


fo 


Q 


fe 


Q 


fc. 


Q 


Lbs. p. 
Sq. In. 


Q 


fc, 


Q 


fo 




Ins. 


Lbs. p. 
Sq. In. 


Ins. 


Lbs. p 
Sq. In. 


Ins. 


Lbs. p. 
Sq. In 


Ins. 


Ins. 


Lbs. p. 
Sq. In. 


Ins. 


Lbs. p. 
Sq. In. 


8,000 


.^•3 


1030 


4.4 


1370 


5-5 


1710 


6.6 


2050 


7-7 


2400 


8.S 


2740 


10,000 


,V7 


920 


4.9 


1220 


6.1 


1530 


7.3 


1840 


8.6 


2140 


9.8 


2450 


12.000 


4.0 


840 


S-A 


1 1 20 


b.7 


1400 


8.0 


1680 


9.4 


i960 


10.7 


2240 


14,000 


4-.^ 


780 


.S.8 


1030 


7.2 


1290 


8.7 


1550 


10. 1 


i8io 


II. 6 


2070 


16,000 


4.6 


730 


6.2 


970 


7 -7 


1210 


9-3 


1450 


10.8 


1690 


12.4 


1940 



In using the preceding formulae it is to be remembered 
that the ordinary unit- of length, as well as the unit of 



Art. 51.] EXACT METHOD OF TREATMENT. 263 

cross-section, is the linear inch, and that the weight i of a 
cubic unit will then be the weight of a cubic inch. This 
investigation will yield results sufficiently accurate for all 
the usual cases of engineering practice, although it does 
not provide for the straightening effect of the pull P, 
except as producing a bending moment oppOvSite to that 
of the uniformly distributed load W. 

Allowance for any other distributed loading than the 
weight of the bar itself, and for any eccentricity of the line 
of action of P that may exist, are made precisely as ex- 
plained in the two paragraphs following eq. (7) of Art. 49. 

Art. 51. — Exact Method of Treating Combined Bending and 

Direct Stress. 

In this method of finding the results of direct stress 
combined with bending it is necessary to determine an 
expression for the centre deflection of the bar, or com- 
pression member, considered as simply supported at each 
end. As the line of action of the direct stress P is sup- 
posed to coincide with the original centre line or axis of 
the bar, if g is the weight per linear unit of the latter, the 
bending moment M^ in the second member of eq. (i), 
Art. 48, becomes 

As this case is one in which P is tension the general 
eq. (i) of Art. 48 will take the following form by the aid 
of eq. (7) of Art. 14: 



a^'w I 



(^x(l-x)-Pw') (i) 



W 
In this equation ^ = -r- is the weight per linear inch. 



264 COMBINED STRESS CONDITIONS. [Ch. VI. 

or Other unit,, of the bar or member producing a bending 
moment opposite to that induced by the direct stress P 
acting with the lever-arm w\ The integration indicated in 
eq. (i) may be completed, but as it is not a simple integra- 
tion it will not be made here. As the greatest bending 
stress is found at the centre of span the centre deflection 
only is needed and a different procedure may be followed. 
Let w^ represent the centre deflection of the member 
considered, a beam simply supported at each end and 
carrying its own weight only, or any other total weight W 
uniformly distributed. It is necessary to use the expres- 
sion for the work performed, or resilience of the beam in 
being deflected at the centre by the amount w^. Eq. (8) 
of Art. 44 gives that resihence as 



WH 



Resiliences^ 7^ (2) 

240/1/ 

In producing the centre deflection n\ the centre of gravity 
of the weight W will descend through the distance w^ found 
by placing Ww^ equal to the resilience given in eq. (2). 
Hence 

WP . . 

240EI ^^^ 

Also, since by eq. (26) of Art. 28 w^ = —x—^y 

Hence the resilience becomes 

Resilience ^Wy^w^. . . . . . (5) 



Art. 51.] EXACT METHOD OF TREATMENT. 265 

If the value of W in teiTns of il\ be taken from eq. (26) 
of Art. 28 and substituted in eq. (5), 

ZT" T 

Resilience =- — ^-^^w,^ (6) 



Hence ike resilience of a bent beam varies as the square 
of the centre deflection. 

If the actual centre deflection of the bar or member 
considered be ■^c'^ the resilience of the beam when deflected 
to that extent will be 

Kesiiience = [ — ^^7 (7) 

\wj 2 4ohI ^'^ 

The curvature of the bar or member being slight, the 
lengths (equal to each other) of the neutral surface with 
the deflections w' and ii\ will be, if V and l^ are the corre- 
spondmg lengths of span or horizontal projections of the 
neutral surface, 



Hence 

'■-'.=K¥-"7^) <" 

The difference r — l^ represents the movement toward 
or from each other of the two ends of the bar or member 
under the action of the direct stress or force P. 

In the case of the eye-bar, the pull of the force P re- 
moves a part of the deflection u.\, and in so doing performs 
work in aiding to lift the weight W of the bar, the remainder 
of the work of Hfting W being performed by the elastic 
efforts of the bar to straighten itself from the deflection 



266 COMBINED STRESS CONDITIONS. [Ch. VI. 

w^ to w\ the latter portion of the work being represented 

by the quantity ^7(1— —"2) • Hence the following 

equation of work may be written, 

The conditions under which the work represented by 
eq. (10) is performed are such that either {2l\ — w') or 
{w^ + w') may be written in the second member. The 
resulting numerical value of w' will be the same in both 
cases but affected by different signs. As the equation is 
written the numerical value of w^ will be negative. 

In eq. (10) there is taken /' =1^=1, the length of panel, 
which may be done with essential accuracy. 

Dividing both sides of eq. (10) by (w^ — w') and solving 
for w\ 

^' = 6Wl^ ^") 

1+ ^ 

25 F Wj 

The deflection w^=-^— ^ appearing m eq. (11) is a 

known quantity. 

After w' is determined, the resultant bending moment 
at the centre of the bar will be 

Wl 
M' = -^-Pw' (12) 

If the area of cross-section of the bar is A, the maxi- 
mum intensity of stress t in it will be, by eq. (2) of Art. 48, 

I /^ M'd\ ' , , 



Art. 51.] EXACT METHOD OF TREATMENT. 267 

Or if the maximum value of t is specified 

, i/^ M'd\ , ^ 

h 
If the section is rectangular, so that A =bh and d = - , 

i = l^ P + -^) ...... (15) 



bh ^ 
and 



When the depth of the bar is small in comparison with 
the length /, it may happen that the resultant or final de- 
flection w' will be such as to make the bending moment 
M' equal to zero. Or 

M'=-j -Pw'=o; .'.w'=jp, . . (17) 

When w' found by eq. (17) is less than w' given by 
eq. (11), eq. (17) is to be employed. This result shows 
that the bar will be subject to no bending, but that it will 
hang like a flexible cable. The conditions thus developed 
are those which indicate when a horizontal or inclined bar 
stressed in tension ceases to act partially as a beam and 
becomes purely or wholly a tie. 

These formulae are perfectly general for all cases of 
bars or members in tension, even for such small sections 
as wire. Their application to individual cases will show 
that excessive intensities will not exist where simple ten- 
sion members are held imder stress in a nearly horizontal 
position. 



268 COMBINED STRESS CONDITIONS [Ch. VI. 



Art. 52. — Combined Bending and Direct Stress in Compression 

Members. 

If the ordinary approximate method of Art. 50 be em- 
ployed, eq. (4) of that article is immediately appjicable, 
using the minus sign in the denominator, P being the total 
direct stress of compression and M^ the bending moment 
due to the uniform transverse load and to eccentricity of 
the line of action of P, if there be any. The greatest in- 
tensity of bendmg stress as represented by that formula 
would then be 

k^f-^ . (X) 

loE 

In this equation, d is the distance from the neutrai axis 
of the section to the extreme fibre in which the intensity k 
exists. 

If e be the eccentricity of the line of action of P, and if 
W be the weight of the compression member whose length 
is /, 

Wl 
M,=^±Pe (2) 

When the moment of P produces bending of the same 
sign with the transverse load W, the plus sign is to be used 
in eq. (2), and the minus sign when those moments are 
opposite. If the line of action of P coincides with the 
axis of the member, the moment Pe disappears from eq. (2). 
Again, if the member is vertical, so that there is no trans- 
verse bending due to the load W, when the line of action 
of P has the eccentricity e, 



M,-^Pe (3) 



Art. 52.] COMBINED STRESSES IN COMPRESSION MEMBERS. 269 

This latter case exists very frequently in the columns 
of buildings. 

Eq. (i) is thus seen to represent the greatest intensity 
of bending stress with AI^ taken from either eq. (2) or 
eq, (3) for the cases of transverse loading, no transverse 
loading, eccentric longitudinal loading, or any combina- 
tion of those cases. 

The resultant intensity of stress, i.e., the greatest 
intensity of compressive stress in the entire compression 
member, will be 

P M^d 

loE 

As A is the area of cross-section, 7=^r^, r being the 

radius of gyration of the cross-section of the compression 

P 
member. If g=-r , eq. (4) will take the form 



(5) 



In the use of this equation, the intensity q must ob- 
viously never exceed the working value given by the column 
formula employed. Indeed, if there is suitable eccentricity 
q may be much less than that working long column value. 

In practical operation the principal use of eq. (5) 
may be the determination of the area of cross-section A 
with some prescribed value of t. It is usuall}^ feasible to 
assign general outside dimensions of the proposed column 
section and that will enable a close approximate value of 
r to be assigned. If, at the same time, an approximate 



p 

1 


M,d 


A 


M^d 


A ' 


Ar^- — ^ 


2 P^' 
loE 



270 COMBINED STRESS CONDITIONS, ]Ch. VI. 

value of q may also be taken, the resolution of the first and 
third members of eq. (5) will at once give 

P P M, d 
A= — F--2 + — ^-2 (6) 

If, on the other hand, such an assignment of q may not 
be made, it will be necessary to solve the first and second 
members of eq. (5), as a quadratic equation, for .4, Bring- 
ing both terms of the second member of eq. (5) over a 
common denominator and solving the resulting equation 
of the second degree in the usual manner, the following 
general value of A will be found: 



y4\ioEr' 



, P MAY P P , , 



t tr^ / loEtr 



h 
Frequently there may be written d=^- and r = .4h. 

Hence 

-2=1 (nearly). 

If, agam, d=- and r = .35/^, 

-2=^ (nearly). 

The preceding values of the radius of gyration r repre- 
sented in terms of the depth h of the compression member 
are closely approximate for practical design work. 

Eqs. (6) and (7) will give the desired area of section of 
the compression member carrying both direct stress and 



Art. 52.] COMBINED STRESSES IN COMPRESSION MEMBERS. 271 

bending produced by transverse loading under the assump- 
tions of the method ordinarily employed. Those formulae 
are sufficiently accurate for their purposes, but it may be 
desirable to use the more exact formulae given in the next 
section. 

Exact Method for Combined Compression and Bending. 

The exact procedure for combined compression and 
bending is identical with that used in Art. 51, the formulas 
determined there simply being adapted to a compressive 
longitudinal force instead of a force of tension. It is to be 
observed, as in the case of the tension member, that the 
compression member may be horizontal or inclined, so as 
to be subjected to bending either from its own weight or 
from some other form of loading in addition to that weight. 
The member may also be subjected to uniform bending 
throughout its length by the eccentric application of the 
longitudinal force P concurrently with the preceding cross 
bending, or, as in the case of a vertical column carrying 
eccentric loading, by that force P alone. 

It is essential to recognize in this connection that while 
the columns may occasionally be in the pin-end condi- 
tion, usually their ends are essentially in a condition of 
at least partial fixedness, although the degree of fixed- 
ness is indeterminate. It will conduce to simpHcity of 
treatment if the transverse bending, either from distributed 
loading or by the eccentricity of application of the column 
load, be treated as if the ends of columns are hinged. It 
has been shown in Art. 28 that the centre deflection of a 
beam of given length and cross-section with ends simply 
supported and with the loading uniformly distributed is 
five times as great as when the ends of the same beam 
are fixed. In the following analysis, therefore, the bend- 



272 COMBINED STRESS CONDITIONS. [Ch. VI. 

ing from both the sources named may be considered as 
produced in a column with hinged ends by a total uni- 
formly distributed load W, sufficient in amount to cause 
one fifth of the actual bending moment acting on the col- 
umn with ends fixed. In this m.anner the fixed or con- 
strained end condition of the actual column is provided 
for, while the simplicity of the hinged end computations 
is retained. The bending moment produced by P, acting 
with the lever-arm of the greatest deflection, will concur 
with the bending moment produced by the own weight 
of the member or other vertical imiform loading, instead 
of being opposed to it, as was the case with the tension 
member of Art. 51. The work performed, therefore, by 
P and the uniform loading W will be equal to the resilience 
or elastic work performed in the member in changing the 
deflection from w^ to w' , it being remembered, in this case, 
that w' may be less than w^. Under these conditions, 
then, eq. (10) of Art. 51, expressing the work done on the 
beam in changing the deflection from the w^ to w' will 
become the following, the second member representing the 
resilience or the work done by the elastic stresses through- 
out its volume : 

Dividing both members of this equation by {w' — w^), 

then solving for w' , the following value of the latter will 

immediately result: 

w. 

-' (9) 



in which 



6_Wl _i__ 
25 P w^ 






Art. 52.] COMBINED STRESSES IN COMPRESSION MEMBERS. 273 

Having found the deflection w\ the general equation 
for the resultant maximum bending moment, eq. (i) of 
Art. 48, will take the following form, in which the coeffi- 
cient c is introduced to provide for fixedness of ends in 
the manner shown in Prob. 4, at the end of this chapter. 
If the ends are hinged, corresponding to the end condition 
of a beam simply supported, c -= i, but if the ends are fixed, 
c may be taken as .5 : 

• M=c(^^±Pie±w')) (10) 

In this equation care must be exercised in using the 
double signs, observing that both plus signs are to be taken 
together as are both minus signs; also, that the eccen- 
tricity ^ in a vertical column is taken in a direction opposite 
to the deflection w^ in which case e is to be considered 
positive and the lever-arm of P is (e-\-w'). In the upper 
chord of bridges ^ may be given such value that 

Wl 
M-=-g—P(^-w') =0 (nearly). . . . (n) 

In the case of vertical columns, like those in buildings, 

Wl . 
ordinarily the term — disappears, leaving the bending 

moment in the column: 

M=P{e + w') (12) 

In the great majority of cases w^ is so small in com- 
parison with e as to make it negligible, so that 

M=Pe (13) 

These various values of the bending moment M cover 
all that usually occur in practical operations. 



2 74 COMBINED STRESS CONDITIONS. [Ch. VI. 

If, in accordance with the preceding notation, t is the 
maximum resultant intensity of stress in the member, 
there will result 

P Md I /^ Md\ 

Evidently the uniform intensity of compressive stress 

P 

^ must not exceed the intensity of working stress given 

by a suitable long column formula. When the greatest 
working intensity t is prescribed, the desired area of cross- 
section of the compression member will be 

--li-^^} ■• (.5) 

d 
The closely approximate values of -^ given immedi- 
ately following eq. (7) may be used in a precisely simiilar 
manner in eq. (15), so as to simplify the practical use of 
that equation. 

Problems for Chapter VI. 

Problem i. — A steel eye-bar 8 ins. by ij ins. in section 
and 32 feet long sustains, in a horizontal position, a tensile 
stress of 144,000 pounds, i.e., 12,000 pounds per square inch. 
Find the greatest bending tensile intensity of stress and 
the resultant intensity of tensile stress at its centre sec- 
tion by the ordinary approximate method of Art. 50, and 
by the exact method of Art. 51. In this problem, £" = 
30,000,000; / = 32Xi2=384 ins.; P = 144,000 and W = 

Ai T 1-5X8x8x8 ^ 
40.8 X 32 = 1306 pounds. Also i = =64. 



Art. 52.] PROBLEMS FOR CHAPTER yi. 275 

By eq. (6) of Art. 50 the resultant intensity of tensile 
stress required is 

1^06X48 

t= 12,000+ ——-, = 12,000+ i860 = 13,860 lbs. per sq. m. 

16 +17 7 ^' ^ ^ 

The centre deflection it\, due to own weight only, used 
in the exact method, is "^'i = .5 inch. Hence, by eq. (11). 
of Art. 51, the centre defirction under tensile stress is 



w^ = '- — 7~ = .i9 inch. 

1 + 1.67 ^ 

The resultant intensity of tensile stress at the centre 
section of the eye-bar, is therefore, : 

A \/ O r- 028 

t = 12,000 ^ ^ , ' ^ o ' == i2,oco +2208= 14,208 lbs. per sq. in. 

1.5X8X8 ^' 

The approximate method, therefore, gives an intensity 
2208 — 1860 = 348 pounds per sq. in. too small. 

Problem 2. — A horizontal square 2 in. by 2 in. steel 
bar 30 ft. long is subjected to a tensile stress of 48,000 
pounds, i.e., 12,000 pounds per square inch. Find the 
same quantities as in Prob. i. £^ = 30,000,000; /=36o 
inches; own weight, ^^ = 408 pounds, and P= 48,000 
poimds. 

By eq. (6) of Art. 35 

^ = 12,000 + 835 = 12,835 lbs. per sq. in. 
In the exact method the centre deflection due to own 
weight is 

Wj = 6 .'2 inches. 

Eq. (11) of Art. 51 gives 

ze;' = 5 . 5 5 inches. 



276 COMBINED STRESS CONDITIONS. [Ch. VI. 

On the other hand, the criterion, eq. (17) of Art. 51, 
gives 

, 408X360 . 

^ =5 .. o — = -3825 inch. 
8X48,000 ^ ^ 

The bar, therefore, will be subject to no bending and 
its stress will be simply that of tension, the centre deflection 
being .3825 inch. If the deflection were sufficient to give 
the bar sensible inclination, it would be necessary to mul- 
tiply the horizontal force P = 48,000 by the secant of that 
inclination to obtain the actual tensile stress in the bar. 

The results given by the ordinary approximate method 
are thus seen to be quite erroneous. 

Problem 3. — A 1.5-inch round steel bar 48 ft. long, carry- 
ing a tensile stress of 10,000 pounds per square inch, is 
inclined at an angle of 51° to the horizontal. Will it be 
subjected to any bending, and what w411 be its centre de- 
flection at right angles to its axis if a = 5 1^? The component 
of the bar's weight producing the deflection named is 
"H^cosa, in which W= 288 pounds is the bar's weight; 
W cos a: = 224 poimds; /=48 ft. =576 ins. By the usual 
formula, w^= 74 ins. Eq. fii) of A'^t. 51 then gives w' 
= 72 ins.; but eq. (17) of Arc. 51 gives 

, 224X576 . , 

w =-7r-z = • 91 mch. 

,8X17,700 

Hence this latter deflection is the true value and the 
bar is subjected to no bending in its stressed condition. 

Problem 4. — A steel column 18 ft. long sustains a load 
of 240,000 poimds and carries a transverse load (i.e., per- 
pendicular to its axis) of 300 pounds per linear foot, the 
latter total being 5400 pounds. The column has a section 
like that shown as "top chord latticed," Page 476, Art. 81, 



Art. 52.] PROBLEMS FOR CHAPTER K/. 277 

and it is composed of two 15-in. by J-in. web plates, two 
3-in. by 3-in. 7 -lb. angles, two 3-in. by 4- in. 14-lb. angles, 
and one i8-in. by to -in. top plate. The sectional area is 35 
sq. ins. The moment of inertia / is 1255, and the radius of 
gyration r is 6. The loading is applied to the latticed 
side of the column, so that the eccentricity of application 
is 8.5 inches. It is required to find the deflection at the 
centre of the column length, the bending moment and 
greatest intensity of stress at the same section. Also, 
if the area of section were not given, find that area if the 
greatest allowed intensity of compression is 12,000 pounds 
per square inch. The details of the column at top and 
bottom are first to be assumed such as to make those ends 
essentially fixed and then hinged. 

If the ends of the column were hinged, the centre bend- 
ing moment w^ould be 

S400X216 ^ ^ ^ . .. 

M = o + 240,000 X 8 . 5 = 2, 185,800 m.-lbs. 

As the ends of the column are first to be taken as fixed, 
and as the deflection in that condition will be but one fifth 
of that existing with ends hinged, it will be necessary to 
take one fifth of the preceding bending moment and place 
it equal to the expression for the centre bending moment 
produced by a imiformly distributed load acting on a 
column supposed to be with hinged ends. If W represents 
that uniformly distributed load, 

-y =437,160 m.-lbs. 

Hence 

ly = 16,200 pounds. 

Byeq. (9a) of Art. 52, £ being 30,000,000 and/ = 2r6 ins., 
w^= .0565 in. 



278 COMBINED STRESS CONDITIONS. [Ch. VI. 

Remembering that P = 240,000, eq. (9) then gives 
w' == .00093 i^- 

These deflections are so small in comparison with 
^ = 8.5 inches, that they will have no sensible effect upon 
the result and they may be neglected. 

In consequence of the elastic motions of the members 
of a steel structure it is difficult to estimate accurately the 
effect of such degree of fixedness of the ends of a column 
as may be attained in an actual structure, but it is probable 
that the resulting bending moment at the centre of the 
column due to eccentricity and lateral loading is not less 
than one half that existing with hinged ends, and that 
ratio will be employed. In eq. (10) of Art. 52, therefore, 
c = . 5 , and the bending moment will be 

M=— ( — ~^ h 240,000 X8.5 j = 1,092,900 in. -lbs. 

Hence by eq. (14) of the same Article the greatest inten- 
sity of compression will be 

240,000 1,092,900 X (^ = 8.5) 

^ 3 5 "12^5 

= 6857 + 7 <^02 =-14,259 lbs. per sq. in. - 

This computation shows the serious effect of eccentric 
application of loading. 

If the greatest allowed intensity of compression is 
12,000 pounds per square inch, eq. (15) of Art. 52 shows 
that the area of cross section required is 



1,092,900 X8.5 
I 240,000 i- 
12,000 



A = ^ ( 240,000 H 7 j =41.3 sq. ms. 



Art. 52.] PROBLEMS FOR CHAPTER VL 279 

The moment of inertia / will now become 1481 instead 
of 1255. 

These results may be compared with those of the ordi- 
nary approximate method by finding the greatest intensity 
of compression, t, by eq. (4) of Art. 52, as follows, after 
displacing r'o by i-z on account of the fixed end condition: 

240,000 , 1,092,000X8.5 

+ — ;T^;—;:r^ =5811 + 7570 



41.3 1481-12 

= 12,135 lbs. per sq. in. 
There is, therefore, no material discrepancy. 

Results corresponding to the preceding, but under the 
supposition that the ends of the column are hinged, may 
readily be found as follows: 

Wl 

-^ = 2,185,800; .'. W = 81,000 pounds. 

Hence 

w^ = .2825 in. 
and 

w' = .0047 i^- • 

While these deflections are five times as large as before, 
nf is still too small to affect sensibly the results and it will 
be neglected. The bending moment at the centre of the 
column will then be 

_^^ 5400X216 

M = ^ g + 240,000X8.5=2,185,800 in.-lbs= 

and the greatest intensity of compression 

240,000 2,185,800X8.5 

35 1255 
= 6857 + 14,800 = 21,657 lbs. per sq. in. 



28o COMBINED STRESS CONDITIONS. [Ch. VI. 

If the greatest allowed intensity of compression is 

12,000 pounds per square inch, the area of cross-section 

becomes 

. I / , 2,185,800X8.5 \ 

A =— 240,000+ 2 =63 sq. ms. 

i2,ooo\ 36 / 

The momicnt of inertia / will now become 2259 instead 
of 1255. 

Comparing these results with those of the ordinary- 
approximate method by finding the greatest intensity of 
compression, t, by eq. (4) of Art. 52, 

240,000 2,185,800X8.5 



63 2259-37 



12,170 lbs. per sq. m. 



This result is a close agreement with the other. 

Problem 5. — The pin of a crank-shaft like that shown 
in Fig. I of Art. 47 sustains a maximum thrust, P, of 
32,000 pounds, the length of crank, e, being 20 inches, 
and the axial distance, /, between the centre of the thrust 
and shaft bearings being 18 inches. Find the diameter 
of the steel shaft at the bearing B if the greatest allowed 
bending tension, k, is 10,000 lbs. per sq. in. and the greatest 
allowed torsional shear, T, is 7000 lbs. per sq. in. 

In using the formulas of Art. 47 the data will be as 
follows : 

^ = 20 ins. ; / = 18 ins. ; AB = 26.9 ins. ; tan j =\^ = i.iii ; 

y = 48°; cos j = .66g\ sin j=.j4i,; P = 32,000 lbs. ; 

^ = 10,000 lbs. per sq. in.; T = 7000 lbs. per sq. in. ; 

bending moment AI = 576,000 in. -lbs. ; twisting mo=' 

ment ilP =640,000 in. -lbs. 

The first method of Art. 47 gives for bending, by using 
the first of eqs. (9), 



^3 [ 32,000 X18 
= 2.i6>J- 



D = 2.i6\J^^-^ =8.34 ins. 

• ^ 10,000 



PART II.— TECHNICAL, 



CHAPTER VIL 

TENSION. 

Art. 53. — General Observations. — Limit of Elasticity. — Yield 

Point. 

Hitherto certain conditions affecting the nature of 
elastic bodies and the mode of applying external forces 
to them, have been assumed as the basis of mathematical 
operations, and from these last have been deduced the 
formulas to be adapted to the use of the engineer. These 
conditions are never realized in nature, but they are 
approached so closely that, by the introduction of empiri- 
cal quantities, the formulae give results of sufficient accu- 
racy for all engineering purposes; at any rate, they are 
the only ones available in the study of the resistance of 
materials. 

In determining the quantity called the " coefficient or 
modulus of elasticity," it is supposed that the body is per- 
fectly elastic, i.e., that it will return to its original form and 
volume when relieved of the action of the external forces, 
also that this "modulus" is constant. There is reason 
to believe that no body known to the engineer is either 
perfectly elastic or possesses a perfectly constant modulus 
of elasticity. Yet within certain limits, the deviations 
from these assumptions are not sufficiently great to vitiate 
their great practical usefulness. 

281 



282 TENSION. [Ch. VII. 

These limits for any given material are in the vicinity 
of the " limit of elasticity " or " elastic limit." The limit 
of elasticity or elastic limit of a material may be defined as 
that point of stress below which the intensity of stress 
divided by the rate of strain, i.e., strain per unit of length, 
is essentially constant. This point or limit is fairly, well 
defined for most grades of structural steel and for some 
other ductile metals, but in other materials like stone or 
timber it is difficult to assign any degree of stress as the 
limit of elasticity. In such material the intensity of stress 
divided by the rate of strain sometimes fails to be constant 
at all. If the intensities of stress and rates of strain for 
such materials be plotted so as to exhibit the relation 
between those quantities the resulting line will be found 
to be a curve without any point which can properly be con- 
sidered the limit of elasticity. Frequently when such 
materials are relieved cf loads, the dimensions of the 
piece subjected to stress will not return to their original 
values. 

Between the extreme limits of these materials exhibiting 
such a range of elastic or physical qualities, all degrees of 
imperfect elastic characteristics may be found. Fortu- 
nately, however, the structural materials commonly em- 
ployed in engineering operations may be treated as if 
possessing at least approximately elastic characteristics 
sufficient to make applicable useful formulae based upon 
Hooke's Law. 

It should be stated that some authorities have given 
arbitrary definitions of the' elastic limit, and that these 
definitions have been much used. Wertheim and others 
have considered the elastic limit to be that force which 
produces a permanent elongation of 0.00005 of the length 
of bar. Again, Styffe defines, as the limit of elasticity, 
a much more complicated quantity. He considers the 



Art. 53.] GENERAL OBSERVATIONS.— LIMIT OF ELASTICITY. 283 

external load to be gradually increased by increments, 
which may be constant, and that each load, thus attained, 
is allowed to act during a number of minutes given by 
taking 100 times the quotient of the increment divided 
by the load. Then the " limit of elasticity " is " that load by 
Which, when it has been operating by successive small incre- 
ments as above described, there is produced an increase in 
the permanent elongation which bears a ratio to the length 
of the bar equal to 0.0 1 (or approximates most nearly to 
o.oi) of the ratio which the increment of weight bears to 
the total load." (Iron and Steel, p. 30.) 

These rather artificial expressions for limit of elasticity, 
however, have now been abandoned in favor of what seems 
to be the most natural value, i.e., the point where the ratio 
between intensity of stress and rate of strain ceases to be 
essentially constant. 

The preceding observations relate to the limit of elas- 
ticity as determined by tests of materials under direct 
tension or compression. Obviously, however, the coeffi- 
cient or modulus of elasticity and elastic limit as well as 
other physical qualities may be determined by subjecting 
beams to flexure. Observed deflections under known loads, 
which do not bend the tested beam beyond the elastic 
limit, will enable the coefficient of elasticity to be com- 
puted by using formulae of the common theory of flexure. 
Similarly the observed increments of transverse loading 
will yield data from which the limit of elasticity may be 
determined. 

By precisely similar procedures the coefficient of elas- 
ticity and elastic limit of material subjected to torsion 
may be found. All such results will be well defined in 
proportion to the elastic properties of the materials. If 
those elastic properties are nearly perfect the results will 
be well defined. On the other hand, they will be obscure 



284 TENSION. [Ch. VII. 

and ill defined if the material possesses only a low degree 
of elastic properties. 

Yield Point. 

In the ordinary testing of materials for engineering pur- 
poses the true elastic limit is not determined. The true 
elastic limit of any test piece is found by carefully com- 
puting the ratio between intensity of stress and rate of 
strain for a loading continually increasing by comparatively 
small increments. Such a procedure is too slow for what 
may be termed the commercial purposes of engineering. 
A much more rapid and convenient procedure consists in 
carefully observing the scale beam of the testing machine. 
As the load is gradually increased the scale beam may 
easily be kept in a horizontal position by moving the scale 
weights until a point of stress in the specimen is reached 
at which the beam drops in consequence of the relatively 
sudden stretching of the material. This stretching con- 
tinues with such a material as structural steel with a slight 
addition of loading, or none at all, to a remarkable extent. 
Finally, after much stretching of the test piece, the strained 
material appears to take on renewed resistance, requiring 
additional loading to produce much elongation. The inten- 
sity of stress in the specimen when this sudden stretching 
begins is called the "yield point" or sometimes the "stretch 
limit." It is but little above the elastic limit. In soft or 
mild steels, or in high structural steel the yield point may 
not be more than two or three thousand pounds above the 
elastic limit. The elastic limit itself is from one-half to 
six-tenths the ultimate resistance for small specimens or 
about one-half the ultimate resistance for large members 
like eye-bars, or a little less than that after annealing. 

The ease with which the yield point may be determined 
has led to its wide use, under the name of elastic limit in 



Art. 54] ULTIMATE RESISTANCE. 285 

much engineering literature, but the distinction should 
always be observed. 

In the case of some structural materials with erratic or 
defective elastic properties, like some grades of cast iron, 
it is practically impossible to find any well-defined elastic 
limit or even yield point. 

Art. 54. — Ultimate Resistance. 

After a piece of material, subjected to stress, has passed 
its elastic limit, the strains increase until failure takes 
place. If the piece is subjected to tensile stress, there 
will be some degree of strain, either at the instant of rup- 
ture or somicw^hat before, accompanied by an intensity of 
stress greater than that existing in the piece in any other 
condition. This greatest intensity of internal resistance 
is called the " Ultimate Resistance." 

In ductile materials this point of greatest resistance is 
found considerably before rupture; the strains beyond it 
increasing rapidly while the resistance decreases until 
separation takes place. 

These phenomena are highly marked in ductile mate- 
rials like wrought iron and structural steel, particularly 
in the latter. In such cases if the application of stress to 
the test piece is carefully controlled a considerable stretch- 
ing of the piece may be produced beyond the point of 
ultimate resistance without actually separating the metal, 
the load per square inch of original section of the piece 
decreasing rapidly. It is not difficult to obtain such re- 
sults with soft or mild steel. 

The ultimate resistances of different materials used in 
engineering constructions can only be determined by 
actual tests, and they hav€ been the objects of many ex- 
periments. 



286 TENSION. [Ch. VII. 

It has been observed in these experiments that many 
influences aft'ect the ultimate resistance of any given 
material, such as mode of manufacture, condition (an- 
nealed or unannealed, etc.)» size of normal cross-section, 
form of normal cross-section, relative dimensions of test 
piece, shape of test piece, etc. In making new experiments 
or drawing deductions from those already made, these and 
similar circumstances should all be carefully considered. 



Art. 55. — Ductility. — Permanent Set. 

One of the most important and valuable characteristics 
of any material is its " ductility," or that property by 
which it is enabled to change its form, beyond the limit 
of elasticit3^ before failure takes place. It is measured 
by the permanent " set," or stretch, in the case of a tensile 
stress, which the test piece possesses after fracture ; also, 
by the decrease of cross-section which the piece suffers at 
the place of fracture. 

In general terms, i.e., for any degree of strain at which 
it occurs, " permanent set" is the strain which remains in 
the piece when the external forces cease their action. It 
will be seen hereafter that in many cases, and perhaps all, 
permanent set decreases during a period of time imme- 
diately subsequent to the removal of stress. Indeed, in 
some cases of small strains it is observed to disappear 
entirely. 

Art. 56. — Cast Iron. 

Modulus of Elasticity and Elastic Limit. 

Cast iron is a metal produced by fusion without sub- 
sequent working such as forging or rolling. Except when 
made for special purposes under conditions of careful control 



Art. 56.] CAST IRON. 287 

of the elements entering it, the quality of the product is 
irregular and variable. Bubbles of gases not escaping from 
the molten mass will leave voids or '* blow-holes " in the 
final product and carbon exists both in the graphitic and 
combined condition, but in varying proportions. The mode 
of production and the practically unavoidable irregularities 
in cooling induce both variable conditions of crystallization 
and internal stresses which are sometimes high enough to 
fracture the completed casting. 

There are some grades of cast iron like those formerly 
used for car wheels and ordnance which give high ultimate 
resistance and comparatively high moduh of elasticity and 
which exhibit an approximation at least to an elastic limit, 
although the latter point is never well defined as in wrought 
iron and steel. The ordinary soft castings used in engi- 
neering practice for water pipes, machine frames and other 
similar purposes disclose under test such erratic properties 
that they cannot be said to have either a well-defined 
modulus of elasticity or any real elastic limit. The irregular 
behavior of cast iron under stress is well shown for different 
grades of the material by the stress-strain diagrams shown 
in Fig. I, in which the vertical ordinates are intensities of 
stress, while the horizontal ordinates or abscissae are the 
strains or elongations per linear inch. These curves are 
typical of what may be considered good grades of cast 
iron for their purpose. The line oq represents a fair grade 
of ordinary soft cast iron, while on and oe belong to a higher 
grade and od a still stronger metal for special purposes. 
The amounts written at the extreme upper ends of the 
curves indicate the loads or stresses per square inch at 
which the test specimens failed. The two curves Of and 
Oe were constructed from data given on pages 597 and 605 
of the '' U. S. Report of Tests of Metals and Other Materi- 
als " for 1899. These two cast-iron test specimens were of 



288 



TENSION. 



[Ch. VII. 



metal of superior or special grades, proposed to be used 
for ordnance purposes, as is indicated by the high ultimate 
resistances, 22,300 and 35,280 pounds per square inch. 

There is seen to be the greatest diversity in the incli- 
nation and general character cf the four strain curves 



35280 lbs. 



32000 



24000 



16000 



8000 




.004 ' PER INCH 



The curve Oe has a fairly straight portion ha, the point a 
representing an intensity of stress of 7000 pounds, while 
the point h represents an intensity of 2000 pounds per 
square inch. The cross-sectional area of this test speci- 
men was I square inch. The difference in strains at the 
two points a and h, orf or a range in intensity of 5000 pounds, 



Art. 56]. CAST IRON-COEFFICIENTS OF ELASTICITY. 289 

was .0003 inch. Hence the coefficient of elasticity for 
these data would be 

• E=-^ — =- 16,667,000 pounds. 
.0003 

In the same manner the increase of strain per linear 
inch of test specimen resulting from increasing the stress 
of 2000 pounds per square inch at k to 8000 pounds per 
square inch at b was .00032. Hence with these data the 
coefficient of elasticity would be 

E = = 1 8, 7 so, 000 pounds. 

.00032 "^ ' ^ 

The strain curve On is an extraordinary one for cast iron, 

as it is straight for nearly its entire length. For the in- 

teUvSity of stress of 16,200 pounds the strain or stretch is 

seen to be .002 inch; hence the coefficient of elasticity 

would be 

77 i6'2oo 

Ji = =8,100,000 pounds. 

.002 ^ 

The metal represented by the strain curve Oq cannot be 
said to have any coefficient of elasticity at all, as no part of 
the curve is straight. These instances selected from a 
large number of tests are representative of what may be 
expected in elastic behavior of cast iron. As a rule, the 
grades possessing the higher ultimate resistances exhibit 
a more nearly normal elastic character and possess what 
may be termed not very well-defined coefficients of elas- 
ticity running from about 14,000,000 to perhaps 18,000,000 
pounds per square inch, while the usual grades or quanti- 
ties employed in engineering castings may have no coeffi- 
cient of elasticity at all or as low as 8,000,000 or 10,000,000 
pounds per square inch. In view of all experimental data 
available at the present time it is probably about as near 



290 TENSION. [Ch. VII. 

correct as practicable to take the tensile coefficient of cast 
iron for ordinary engineering purposes, as 

£=-12,000,000 to 14,000,000 pounds, 

or one half that of wrought iron. For the special grades 
of stronger cast iron, such as are used for ordnance and 
car-wheel purposes, a coefficient or modulus of 16,000,000 
pounds to 18,000,000 pounds per square inch may be used. 
As is usually the case in cast iron, the elastic limits of 
the curves in Fig. i are so ill-defined that they cannot be 
placed with certainty even on the curves Of and Oe, or 
scarcely on On, and not at all on curve Oq. If the points 
are approximately located on the first three of these curves 
they may perhaps be placed at b (8000 pounds per square 
inch), at a (7000 pounds per square inch), and at m (19,000 
pounds per square inch). In none of these cases, however, 
can the metal be said to have either a well-defined limit of 
elasticity or a true yield point, and that observation is in 
general true of all cast iron. 



Resilience, or Work Performed in Straining Cast Iron. 

As the scale of the original of Fig. i was 8000 pounds 
to each inch of vertical ordinate and .001 inch to each inch 
or horizontal ordinate or abscissa, and as the strains shown 
in Fig. I belong to a test piece i inch square in section and 
I inch long, each square inch of area on the original dia- 
gram between any one of the strain curves and the axis 
of abscissae drawn through will represent 8000 X. 001 =8 
inch-pounds of work performed in stretching that test 
piece. The strain at the point h on the curve Of is .00036 
inch, as shown in the figure, while the mean intensity of 
stress in producing that strain is 4400 pounds. Hence if 



Art. 56.] CAST IRON.—RESILIENCE. 291 

b represents the elastic limit the resilience or work per- 
formed in stretching the metal up to the elastic limit of 
8000 poimds per square inch is 

4400 X .00036 = 1.58 inch-pounds per cubic inch. 

Similarly, if a is the elastic limit in the strain curve Oc, 
the total strain for each inch in length of the test specimen 
is .00038 inch and the mean intensity of stress is 3750 
pounds, all as shown in Fig. i. Hence the resilience or 
work performed was 

3750 X .00038 = 1.43 inch-pounds per cubic inch. 

A similar computation may be made for the straight por- 
tion of the strain curve On, but the preceding operations 
sufficiently illustrate the procedure. 

The total work performed in breaking each specimen 
may readily be found in precisely the same manner. In 
the case of the curve Of the strains or elongations of the 
specimen were actually observed only up to the point d, 
although failure actually took place at / or at the intensity, 
35,280 pounds per square inch. The part df of the curve 
is drawn approximately as a continuation of the observed 
curve and therefore is shown as a broken line. The area 
included between the curve Of and the horizontal ordinate 
Os, i.e. the area of the figure Ofs, is 11.97 square inches. 
Hence the work performed in rupturing the test piece was 

11.97 X 8000 X. 00 1 =95.76 inch-pounds per cubic inch. 

Again, in the case of the strain curve Oe the area of the 
figure Oct is 4.69 square inches. The total work expended, 
therefore, in rupturing the specimen was 

4.69 X8000 X .001 =37.5 inch-pounds per cubic inch. 

In the latter case the short portion ce of the strain curve is 



292 TENSION. [Ch. VII. 

drawn approximately, as the strain observations ceased 
at c. It is to be remembered, as is indicated in each of 
these cases, that when the data apply to each linear inch 
of test piece and each square inch of sectional area, the 
work computed will be for i cubic inch of material. It 
is only necessary to multiply by the number of cubic inches 
in the test piece in order to obtain the work performed in 
the entire piece. 

Ultimate Resistance. 

The ultimate tensile resistance of cast iron is an ex- 
ceedingly variable quantity; it may range from not more 
than 8000 or 10,000 pounds in castings of indifferent quality 
to values of nearly 50,000 pounds per square inch in such 
special grades of metal as those which have been used for 
car wheels and ordnance. Cast iron has passed com- 
pletely out of use for the manufacture of heavy guns, but 
there are other ordnance purposes for which it is still 
used. The castings usually employed by civil engineers 
are generally of soft-grade iron; they afe such as water 
pipes, frames, beds of machines, and other similar purposes 
which do not require special grades produced by special 
mixtures of raw material or special processes of manu- 
facture. The ultimate resistances will, therefore, be con- 
siderably less than those belonging to ordnance and car- 
wheel irons, or for specially strong grades of metal. As 
with all material, the character of cast iron affects to a 
great extent its resistance, i.e., w^hether it is fine or coarse 
grained, as does also the character of the ore from which 
it is produced. 

Three specimens turned down to a diameter of about 
.625 inch taken from iron used in the Boston water pipes 
and broken at the .Warren Foundry, Phillipsburg, New 



Art. 56.] 



CAST IRON. 



293 



Jersey, gave the following ultimate resistances in pounds 
per square inch: 



18,300, 



15^470, 



13,070. 



These results represent fairly the ultimate resistance of 
ordinary cast-iron pipe and other castings commonly used 
in civil engineering practice. It has sometimes been stated 
that the outer surface or " skin " of iron castings has a 
greater capacity of resistance to stress than the interior 
parts. Investigations carefully conducted, however, by the 
late Professor J. B. Johnson and others do not show that 
to be the case. Indeed it is practically certain that there 
is no essential difference bet^veen the resistances of the 
exterior and interior parts of a casting unless it has been 
subjected to some special treatment. It is not unlikely 
that this erroneous impression may have arisen from the 
results of irregular cooling of castings producing internal 
stresses sometimes sufficient to produce fracture. 

The ''Report of the Tests of Metals and Other Mate- 
rials" at the United States Arsenal, Watertown, Mass., 
for 1900, contains a mass of tensile tests of pig irons and 
ordnance castings of a great variety of grades and quali- 
ties, from which the following tabular statement of greatest 
and least values have been taken. There are also given 
the results of -two tests of gear teeth taken from the sam.e 

source. 

TENSILE TESTS OF CAST IRON. 



Iron. 


Ultimate Resistance. Lbs. per Sq. In. 


Greatest. 


Least. 


Pig 

Ordnance 

Castings . 
Gear teeth. 


31,890 Fine granular, gray, 

33,500 " 

12,200 Fine or medium granu- 
lar, gray. 


1 1,820 Coarse granular, dark gray. 

14,900 " " " " 
12,080 Fine or medium granu- 
lar, gray. 



294 TENSION. [Ch. VII. 

As a recapitulation there may be written: 
For ordinary castings: 

[ 12,000,000 lbs. per sq. in. 
Modulus of elasticity -. to 

( 14,000,000 lbs. per sq. in. 

Ultimate tensile resistance, 15,000 to 18,000 lbs. per sq. in. • 
For specially excellent grades: 

t 16,000,000 lbs. per sq. in. 
Modulus of elasticity } to 

( 18,000,000 lbs. per sq. in. 
Ultimate tensile resistance, 20,000 to 35,000 lbs. per. sq. in. 

Tensile working resistances in pounds per square inch 
may be taken as follows : 

For water pipes and other similar purposes : 

3000 to 3500 lbs. per sq. in. 

With higher grades of cast iron for special purposes: 

4000 to 7000 lbs. per sq. in. 

Ejects of Remelting, Contiraied Fusion, Repetition of Stress, 
and High Temperatures. 

The physical qualities of cast iron may be much im- 
proved by remelting and continued fusion. The product 
of the blast furnace is commercial pig iron. These pigs 
remelted, as in a cupola furnace, form the ordinary cast- 
ings of engineering work. If this remelting should be con- 
tinued so as to secure third or fourth fusion metal the 
resisting properties of the iron would be enhanced, but the 
cost would at the same time be materially increased, and 
hence second fusion metal only is ordinarily used. 

Again, experience has shown that if molten metal be 
held in fusion, even for a period of three hours or more, 
its physical quality continues to improve, but the cost of 



Art. 57.] WROUGHT IRON.— MODULUS OF ELASTICITY. 295 

such a procedure renders it prohibitive for ordinary pur- 
poses. 

Many investigations have been made to determine the 
resisting power of structural materials to frequent and con- 
tinued repetition of stresses, not only below, but above the 
elastic limit, the relief from stress between two applications 
sometimics being partial and sometimes complete. It has 
been found that such repeated stresses, when as high as 
one-half to three-quarters of the ultimate resistance, pro- 
duce material fatigue in cast iron and final failure much 
below the ordinary ultimate resistance as determined by a 
gradual application of load. Such tests have shown that 
cast iron is somewhat more sensitive to fatigue than the 
ductile structural materials of higher ultimate resistance. 

The effect of high temperatures upon the resisting 
capacity of cast iron is not in general different from that 
found for steel and wrought iron. Little, if any, softening 
is observed until a temperature of 500° F. is approached, 
but beyond that limit it is liable to begin to lose capacity 
of resistance to a material extent if not rapidly. 

Art. 57. — ^Wrought Iron. — Modulus of Elasticity.^Limit of Elas- . 
ticity and Yield Point. — Resilience. — Ultimate Resistance and 
Ductility. 

Wrought iron as a structural material has been com- 
pletely displaced by the various grades of structural steel, 
although it is still used in relatively small quantities for 
special purposes. Again, many bridge and other struc- 
tures built of wrought iron are still standing, and it is 
essential to retain a record of its physical qualities. 

Wrought iron differs fundamentally from steel in its 
manner of production, as it is a product of the puddling 
furnace. A white-hot spongy mass was brought out of a 
bath of molten slag and passed between rolls, resulting in 



296 



TENSION. 



[Ch. VII. 



what were known as puddle bars. These were cut in suit- 
able lengths, and placed in rectangular packages or piles 
of proper size to produce the finished bar or beam by sub- 
sequent heating and rolling. 

This process of production gave to wrought iron a 
fibrous internal structure of much greater ultimate resist- 
ance in the direction of the fibre than at right angles to 
the fibre or direction of rolling, and this was true whatever 
shape was produced, such as plates, beams, bars, etc. 

Modulus of Elasticity. 

The coefficient or modulus of elasticity of wrought iron 
was determined by many tests of both small and full-size 
bars when it was the principal structural material in bridges 
and other similar structures. The adjoining table gives 
the results of tests of four bars only. The two i-inch square 
bars were of fine quality of wrought iron and were tested 
many years ago by Eaton Hodgkinson. The results of 
tests of the 5 -inch and 3 -inch bars are taken from the 
" Report of Tests of Metals for 1881 " made on the large 
testing machine at the U. S. Arsenal, Watertown, Mass. 
The table below gives full information as to the total strain, 
gage length and stress per square inch for the various bars. 
If p is the stress per square inch and / the strain per linear 
inch of gaged length, the coefficient of elasticity E will have 
the value. 



Size of Bar. 
Inches. 


Lenlt Total Strain. 
Inches. | Inches. 


Stress per 
Sq. Inch, 
Pounds. 


E. 


iXi 
iXi 

504X1. 27 
305x1 


120 

120 

80 

80 


•04556 
■043 
.029 
.0279 


10,670 
10,095 
20,000 
20,000 


28,101,000 
28,198,000 
27,586,000 
28,674,000 



Art. 57] L/M/r OF ELASTICITY AND YIELD POINT. 297 

It will be observed that the four values shown are more 
nearly the same than will be found in a long series of deter- 
minations in the early tests of engineering materials when 
wrought iron was in general use. As a result of such 
determinations, the value of 

E = 26,000,000 

may be taken as a fair average value for wrought iron 
members of structures. For small specimens, or for some 
special grades of wrought iron, 27,000,000 or possibly 
28,000,000 may be used. 

Obviously all values of E must be computed for inten- 
sities of stress less than the elastic limit. 

Limit of Elasticity and Yield Point — Resilience. 

The limit of elasticity for wrought iron is not nearly so 
well defined as for structural steel. The diagram Fig. i has 
been constructed from the test of the one-inch square 
wrought iron bar with a gaged length of 10 feet and with a 
load increasing by small increments. The horizontal ordi- 
nates represent the total strains in inches, while the ver- 
tical ordinates represent intensities of stress per square 
inch. 

That part of the curve- from the origin o to a is straight 
and its equation is, 

p=El. 

Above a the line begins to curve and at e the curvature 
becomes about as sharp as at any point. The point a, 
elastic limit, may be taken at 26,000 pounds per square 
inch, while e, the yield point, may be considered as 29,000 
pounds per square inch, although this latter point is not 
well defined. Above e the curve becomes much less inclined 



298 



TENSION. 



[Ch. VII. 



to a horizontal line, showing that for small increments of 
load the stretch of the specimen is relatively great. 

While these results belong to one specimen only of 
wrought iron they are characteristic of the metal. Approxi- 
mately the elastic limit may be considered half the ultimate 



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Fig. 



resistance and the yield point possibly 2000 to 4000 pounds 
more. 

For all ordinary cases of wrought-iron structures the 
elastic limit may safely be considered 22,000 to 24,000 
pounds per square inch and the yield point from 25,000 to 
28,000 pounds per square inch, as it is to be remembered 
that the elastic limit and yield point will be higher for 
small test specimens than for full-size structural mem- 
bers. 



Art. 57.] WROUGHT IRON.— DUCTILITY AND RESILIENCE. 299 

Ductility and Resilience. 

In Fig. I the horizontal coordinates of the stress-strain 
curve are the strains for 120 inches in length of a wrought- 
iron test bar, corresponding at each point to the intensities 
of stress per square inch shown on the vertical line through 
0. This curve exhibits fully the physical characteristics of 
the material under test. The straight part oa of the curve 
belongs to that part of the loading below the elastic limit 
a, i.e., below 26,000 pounds per square inch. The point e 
indicates the stretch limit at about 29,000 pounds per 
square inch. There is no constant proportionality be- 
tween stress and strain above a nor is there any great 
increase in the strain for a given small increment of load- 
ing until the point e is passed, but above that point the 
stretch for each constant increment of loading becomes 
relatively large. Beyond the point b, the inclination of 
the stress-strain curve to horizontal is relatively small. At 
or near c the curve becomes horizontal, showing the maxi- 
mum intensity of resistance, i.e., the ultimate resistance, 
and the broken line cf indicates a rapidly decreasing load 
if the testing machine is properly manipulated prior to the 
actual parting of the material at /. Usually the actual 
failure of the material will take place at the highest point 
of the curve unless special pains be taken to operate the 
decrease of loading and even under such conditions the 
material must be highly ductile to produce the part of 
the curve shown by the broken line. 

The resilience of w^ork expended below the elastic limit 
a can readily be computed by the aid of Fig. i, as it is 
represented by the triangular area between the straight 
part of the stress-strain curve and a vertical line through 
its upper limit. The strain at the elastic limit of 26,000 
pounds per square inch is .11744 inch. The average force 



300 TENSION. [Ch. VII. 

acting upon the specimen up to the elastic hmit would be 
half the value of the latter. Hence the elastic resilience 
or the work performed on the specimen up to the elastic 
limit is 

26,000 . . . 

.11744X = 1527.6 mch-pounds. 

Inasmuch as the test specimen was 120 inches long, the 
elastic resilience of the bar would be 12.73 inch-pounds 
per cubic inch of its volume. Similarly, the area of the 
irregular figure oebch is 4.97 square inches, and as the 
scale of force is 20,000 pounds per linear inch, that figure 
represents 4.97X20,000 pounds =99,360 inch-pounds of 

work; or — '~ =828 inch-pounds of work per cubic inch 

of volume of the test specimen. If this test-bar, therefore, 
were to be broken by a falling weight of 100 pounds, that 
weight would be required, to fall through a height of 

99,360 ^ • -, 

— =993.6 mches. 

100 

It is clear from the figure that if the metal possessed 
little ductility so that its strain curve extended no further 
than the point b, the w^ork required to be expended in 
breaking it would be very small compared with that needed 
for luipturing the actual wrought -iron piece. The effect 
of a falling w^eight may represent a shock or blow^ or be 
taken as the equivalent of what is usually called a suddenly 
applied load. These considerations show w^hy a ductile 
material requiring so much more work to be performed 
to break it is much better adapted to sustain shock than 
a non-ductile or brittle material. The latter class of 
materials can be strained so little before failure that little 
work is required to be expended to break them. 



Art. 57.] WROUGHT IRON.— ULTIMATE RESISTANCE. 301 

Ultimate Resistance. 

The ultimate resistance of wrought iron depends to 
some. extent, like structural steel, on the size of the test 
specimen or bar, its treatment during manufacture, and 
whether the piece is tested in the direction in which it was 
rolled or at right angles to that direction. Wrought iron 
being a fibrous material, its ultimate resistance is materially 
greater in the direction of the fiber than at right angles to 
that direction or in inclined directions. Structural speci- 
fications usually prescribed that when used in tension, 
wrought iron should take its load parallel to the direction 
of rolling, particularly for wrought-iron plates. 

Round and rectangular bars of wrought iron of ordinary 
structural sizes showed under tests ultimate resistances, 
generally varying from about 45,000 to 50,000 pounds per 
square inch, the smaller values applying to large bars and 
the large values to bars of small section. 

A series of tests of round bars found in the '' Report of 
the Committee of the U. S. Board Appointed to Test Iron, 
Steel, and other Metals, etc.," showed that the ultimate 
resistances ran from about 60,000 per square inch for J-inch 
rounds down to about 46,000 to 47,000 pounds per square 
inch for bars 4 inches in diameter. 

The ultimate resistance of such wrought-iron shapes as 
angles, eye bars, channels, tees, and others were shown by 
many tests to be about the same as bars and flats of the 
same quality and size, i.e., many test specimens showed 
ultimate resistances running from about 45,000 to 50,000 
pounds per square inch. If the shapes or plates were 
small, so that the temperature was relatively low during 
final passes between the rolls, the hardening effect of such 
treatment would raise the ultimate resistance to some 
extent, resulting in higher values than for similar shapes 



302 TENSION. [Ch. VII. 

of large section which suffered less reduction of temperatures 
during the process of rolling. Thin plates showed markedly 
higher ultimate resistances than thick plates for this reason. 

Ductility. 

From what has been stated it is evident that wrought 
iron would show the greatest final contraction of fractured 
area and final stretch when tested in the direction of rolling 
than in any other direction. Again it is equally clear 
that the percentage of final stretch would be materially 
greater for short specimens than for long ones, because the 
necking-down at the section of fracture would add a much 
greater percentage to the length of a short specimen than 
to a long one. While both final contraction and final 
stretch varied greatly in different test pieces, it may be 
stated that for gage lengths ranging from about 5 feet 
to 20 feet, full-size wrought-iron bars gave a final con- 
traction of 20 per cent to 30 per cent and a final stretch 
of about half these values. 

Test specimens of plates, angles and other shapes, the 
final stretch being measured over a gage length of 8 inches, 
would generally 3deld about 20 per cent to 30 per cent of 
final contraction and about 10 per cent to 20 per cent of 
final stretch. 

The preceding ma}^ be considered fairty representative 
values of ductility of the best quality of wrought iron used 
in bridge and other structures. They show that the metal 
was highly ductile and well adapted to structural purposes, 
although possessing these desirable qualities to a less degree 
than structural steel. 

Fracture of Wrought Iron. 

The characteristic fracture of wrought iron broken in 
tension either directly or transversely is rather coarsely 



Art. 58.1 STEEL— MODULUS OF ELASTICITY:^ 303 

fibrous, not infrequently exhibiting a few bright granular 
spote, which, in rare cases, may possibly be crystalhne. 
This characteristic fibrous fracture is produced by the 
steady application of load, but a piece of wrought iron 
will exhibit a granular fracture if broken suddenly. Many 
statements have been made that wrought iron may become 
crystalline and lose both ultimate resistance and ductility 
under certain conditions of use, but bright granular fracture 
has probably been mistaken in such cases for crystalline. 

Art. 58.— Steel. 

Modulus of Elasticity. 

The great number of varieties and grades of steel brings 
into existence a correspondingly great number of physical 
quantities and coefficients or moduli used in its consider- 
ation in connection with the '' Resistance of Materials." 

Notwithstanding the number of varieties of steel used 
at the present time for engineering purposes, it is fortunate 
in the interests of simplified computations to find their 
moduli of elasticity varying so little that they may be 
taken as practically the same. Again, it is further fortunate 
that the moduli for tension and compression also appear 
to be the same, and they are so taken. 

That class of steel generally to be considered here 
is included under the term ''Structural Steel," which 
may be divided into low, medium, and high steel. These 
three grades of structural steel are mainly based upon the 
amounts of carbon which they contain. While each class 
shades insensibly into another without well-defined limits, 
it may be approximately stated at least that low or soft 
steel will have carbon ranging from about .1 to .2 per cent., 
and that the carbon in medium steel will run from about 
.2. to .3 per cent., while high steel will show about .3 to .45 



304 TENSION. [Ch. VII. 

per cent, of carbon. The ultimate resistance of low steel 
may run from 52,000 to 60,000 pounds per square inch, 
medium steel from 60,000 to 68,000 pounds per square 
inch, and high steel from 68,000 to about 76,000 pounds 
per square inch, or possibly higher. Experimental inves- 
tigations have shown that the coefficient of elasticity is 
essentially the same for all grades of steel used in construc- 
tion. This observation holds true also for nickel steel, 
which has within the past few years come into use for 
special structural purposes. A considerable number of 
tests of nickel-steel specimens, in some cases containing 
3.375 per cent, of nickel with .3 per cent, of carbon and .73 
per cent, of manganese, given in the U. S. Report of Tests 
of Metals for 1898 and 1899, show that the coefficient of 
elasticity for this metal may be taken at values ranging 
from 28,700,000 pounds to 30,385,000 pounds per square 
inch. In other words, the coefficient of elasticity of this 
nickel steel may be taken between the usual limits for 
ordinary structural steel of 28,000,000 and 30,000,000 
pounds per square inch. 

Table I gives a condensed statement of the results of 
an extended investigation made to determine the " con- 
stants " of structural steel by Prof, (now President) P. C. 
Ricketts, at the mechanical laboratory of the Rens. Pol. 
Inst, in 1886. Although these tests were made before as 
many varieties and grades of steel had been developed as 
at present, the values given in the table are accurately 
characteristic of the same grades of structural steel pro- 
duced at the present time, 191 5. As no corresponding 
determinations have been made of such wide range nor with 
such a wide scope of purpose since that early date, the 
table has unique value and is worthy of careful study. 
Although this table contains other values than those im- 
mediately desired, the opportunity of directly comparing 



Art. 58. 



STEEL. 

Table I, 



305 







Per 

Cent. 


TKN'SION 




Specimen 




PoLin 


dsper Square Inch. 




Mark. 


Car- 
bon. 




















Per 


Per 




Ulti- 










Diam. 


Cent. 


Cent. 


Elastic 


mate 


Coefficient 








Inches. 


Reduc. 


ElonR 


Limit 


Resist- 


of Elas. 










of Area 


in 81ns. 




ance. 




Rivet steel * . . . 


Tl 


.09 


0.756 


61.7 


30.5 


39.600 


63,600 


30,039,000 








0.758 


61.7 


30.5 


38,800 


63,300 


30,010,000 




13 


" 


0.757 


60.8 


28.9 


37.800 


63,000 


31,160,000 




41 


" 


o. 757 


65.3 


29.6 


37,800 


62.000 


31,063,000 




42 


' ' 


0.75S 


65.1 


29.4 


38,600 


63,200 


30,471,000 




43 


' ' 


0.758 


62.3 


29.9 


39,400 


62,800 


29,9(35,000 




61 


' ' 


0.760 


61 .6 


30.1 


37,400 


60,600 


30,456,000 




6,. 




0. 760 


60.6 


29.6 


36,900 


61,300 


30,885,000 




63 


' ' 


0. 760 


61.8 


32.2 


39,100 


61,900 


27,335,000 




■81 


' ' 


0. 760 


57.9 


29. 2 


38,100 


62,500 


30,618,000 




82 


' ' 


0. 7 50 


62.4 


28.4 


37,100 


62,300 


30,172,000 




83 


" 


0.758 


61 .0 


28.2 


36,600 


61,400 


30,424,000 




101 


' ' 


. 756 


65.7 


28.6 


35 600 


61,700 


29 696, qoo 




1 02 


' ' 


0.755 


64.7 


29.0 


36,800 


61,600 


30,075,000 




I Oh 


' ' 


0.754 


64-3 


29.1 


36,900 


62,100 


30,37 1 .000 




31 




0.757 


63.4 


27.9 


36,700 


61,200 


30,918,000 




32 


' ' 


0.758 


64.0 


30.4 


37,700 


61,900 


30,801,000 




33 


' ' 


0.758 


64.3 


29.2 


37,100 


61,800 


31,091 ,000 




51 


' ' 


0.757 


51.7 


30.1 


37,800 


62,900 


30,032,000 




52 


' ' 


0.755 


49-4^ 


29.2 


38,500 


63,600 


31,646,000 




53 


' ' 


0.757 


51-2 


28.1 


37,800 


61,300 


30,031,000 




71 


' ' 


0.750 


62.1 


30.9 


36,200 


6t,2oo 


30,166,000 




72 


" 


0.749 


60.5 


29.6 


36,800 


62,400 


30,415,000 




73 


" 


0.751 


61.3 


31-7 


37,800 


62,000 


30,232,000 




III 


II 


0.752 


64.3 


29.4 


36,400 


62,400 


30,030,000 




II2 


" 


0.7S4 


63.0 


29.4 


36,400 


61,700 


30,556,000 




113 


II 


0.749 


62.3 


29.2 


•36,700 


62,200 


30,011,000 




2l 




0.752 


55-1 


29.9 


37,200 


61,600 


30,210,000 




22 




0-757 


53-7 


31.0 


36,700 


60,100 


32,965,000 




J^ 




0.753 


53-2 


32.0 


39,300 


61,000 


30,097,000 


Bessemer t • • • 


Ni 


.11 


0.748 


60.3 


28.4 


41,500 


66,600 


28,950,000 




N, 

^^ 
Oi 

1 
&i 
U2 
U3 

I2 

?? 


II 
" 


0.754 


58.3 


28.2 


41,400 


65,200 


29,391,000 






0. 750 


57.0 


28.2 


43,400 


67,000 


29,809,000 




. 12 


0.751 


59.7 


27.4 


41,500 


65,300 


29,186,000 




II 


0. 750 


59.2 


28.5 


41,100 


65,100 


29,252,000 






0. 750 


57.4 


27.0 


41,400 


65,700 


29,464,000 






0.747 


57-3 


30.6 


42,000 


66,100 


29,007,000 




, , 


0. 750 




30. 1 


41,900 


65,400 


29,809,000 






0.751 


57 ■ I 


28.7 


41,300 


65,400 


20.270,000 




■J-^3 


0.763 


58.1 


26.8 


48,100 


69,400 


20,706,000 






. 760 


59.5 


27 .0 


47.400 


69,300 


20,500,000 




II 


0. 760 


56.4 


27. T 


47,100 


70,100 


20,238,000 






. 763 


SO. I 


28.2 


42,200 


65,300 


29,430.000 






O; 760 


=;6.6 


27.6 


42,300 


65,600 


20.678,00c 






0.756 


58.3 


27 .0 


42,300 


66,400 


29,300,000 




.t6 


0.747 


54.8 


28.9 


42,000 


68,^00 


30,083,000 




II 


0.745 


5 5-7 


27.6 


41,700 


68,500 


30,26(5,000 






0.745 


5 5 • 


27.4 


41,000 


68,600 


20,442,000 




\V 


0.746 


56.3 


27.1 


42,100 


70,400 


20,375,000 






0.744. 


57-2 


27.4 


42,700 


70,500 


30,158,000 




?x^ 




0.749 


55.8 


27.1 


41,500 


79,600 


30.784,000 




X' 


.^6 


0. 761 


40.7 


20.5 


60,900 


97,500 


29,045 000 




7r^ 


" 


0.756 


38.5 


19. I 


60,400 


99,600 


30,236,000 




Vo 




0.759 


39-5 


19.4 


69,700 


99,100 


20,080.000 




Sri 


• 39 


0.763 


39-0 


20.0 


69.500 


95,800 


30,025 000 




Jr' 


' ' 


. 762 


36.8 


19.2 


69,600 


96,200 


30,044,000 




W3 




0.765 


36.7 


19.0 


69,100 


95, 200 


20,291,000 



* Open hearth front "^tprUon, Pa. 



t Prom Troy, N. Y. 



3o6 



TENSION. 
Table I. — Continued. 



[Ch. VII. 









SHEAR. 






cox 


PRESSION. 






















Double 




■ 


P 


ounds per Square Inch 




Shear 






Ultimate 










1 


Over 


Pounds per Square Inch. | 


Single 


Shear. 


Double 


Shear. 


Single 




1 








1 


Shear 
Ultimata. 


■"" 










I 


Elastic 


Coefficient of 


Elastic 


Ultimate 


Elastic 


Ultimate 




Limit. 


Elasticity. 


Limit. 


Resist. 


Limit. 


Resist. 




39,000 


29,897,000 








; 




3y,5oo 


27,113,000 


39,600 


4 1,440 


43,600 


46,460 


1 .022 


30,000 


28,444,000 












41,100 


29,110,000 












41,100 


29,025,000 


34.600 


45,260 


38,200 


47,450 


1.048 


41,000 


29,045,000 












40,200 


30,045,000 












40,200 


28,853,000 


31,500 


46,020 


33,800 


47.590 


1.034 


40,400 


29,411,000 












41 ,600 


30,192,000 












41,600 


29,302,000 


31.700 


46,910 


33,500 


48,390 


1 .032 


41,600 


29,216,000 












38,600 


29,013,000 












38,600 


29,963,000 


31,100 


44,780 


34,000 


46,590 


1 .040 


38,600 


29,478,000 












38,300 


20,090,000 












38,300 


29,807,000 


35,900 


44,600 


38,500 


47,350 


1 .062 


38,300 


28,961,000 












41,700 


29,630,000 












41,700 


28,941,000 


33,800 


46,440 


39,400 


48,890 


1.053 


41,700 


29,696,000 












30,QOO 


29.437,000 












40,000 


30,009,000 


33,700 


45.190 


35,700 


47,210 


1.045 


40,000 


28,730,000 












30,500 


29,005,000 












39,700 


29,740,000 






. 






39,900 


29,963,000 












40,000 


31,433,000 












40,000 


29,782,000 


35,800 


46,100 


40,700 


47.210 


1 .024 


39,700 


29,391,000 












41,800 


28,567,000 












41,700 


29,144,000 


30,500 


49.210 


38,600 


51,000 


1 .036 


41,700 


28,747,000 












41,100 


28,503,000 












41,400 


29,531,000 


34,400 


51,470 


39,500 


51,470 


1. 000 


41,200 


28,730,000 












42,600 


29,162,000 












42,400 


29,210,000 


37,000 


49,740 


40,300 


50,940 


1.024 


41,900 


28,635,000 












44,400 


28,070,000 












44,800 


28,729,000 












45,000 


29,025,000 












44,100 
44,300 


29,281,000 
29,830,000 


36,600 


51,000 


40,800 


51,510 


1 .010 


44,200 


29,324,000 


i 










4T,IOO 

41,400 


28,812,000 


1 










29,342,000 


36,700 


51,280 


43,800 


52,550 


1 .025 


41,000 


28,666,000 












41,400 
41 ,600 


28,860,000 












29,241,000 


41,50* 


53,260 


46,000 


53,390 


1 .002 


41,800 


29,802,000 












55.200 


29,162,000 












54.400 


29,454.000 


52,500 


70,190 








54,400 


29,281,000 












50.500 


28,602,000 












59,200 


28,981,000 , 


51,900 


67,760 








59,500 


29,281,000 













Art. 58.] STEEL. 307 

different ph3^sical constants from the same quality of steel 
is a sufficient reason for inserting the entire table at this 
place. All the test pieces were uniformly about three- 
quarters of an inch in diameter, and the stretch was in all 
cases measured on 8 inches. The elongations given are per 
cents of the original length of 8 inches. 

The reductions of area are the per cents of original 
sections of the test pieces which indicate the differences 
between the original and fractured areas. 

As indicated, the first half of the table belongs to speci- 
mens of open-hearth rivet steel from Steelton, Pa., while 
the second half contains results draw^n from tests on a com- 
paratively wide range of metal from the Bessemer process 
of the Troy Steel and Iron Co., of Troy, N. Y. The open- 
hearth rivet steel is all seen to contain only .09 per cent, 
of carbon, while the Bessemer metal had carbon varying 
from 0.1 1 per cent, to 0.39 per cent., with a wide gap 
between 0.17 and 0.36 per cent. 

The specimens i^, 1^, and I3 were cut from the two ends 
and centre of bar i, and those subjected to tension were 
located adjacent to specimens of the same name subjected 
to compression. Similar observations apply to other sets 
of specimens affected by the same figure or same letter. 
Hence there is shown in this table the relation of different 
physical quantities belonging to as nearly identically the 
same material as the possibilities of the case admit. 

The coefficients of tensile elasticity exhibit unusual 
uniformity. Those for the open-hearth steel show no 
variation with the small variation in carbon. Although 
the tensile coefficients for the Bessemer steel are slightly 
lower for the lowest per cents of carbon than for the highest, 
yet some of the lowest coefficients are found for the highest 
carbons, and it is difficult to determine any essential varia- 
tion with varying proportions of that element. 



3o8 TENSION. [Ch. VII. 

While the average of the tensile coefficients is a very 
little more for the open hearth than for the Bessemer steel, 
there is really no sensible difference between them. The 
average tensile coefficient may be taken at 30,000,000 
pounds per square inch. 

Too much importance should not be attached to the 
percentage of carbon alone in these specimens, as "the 
presence of other elements not given, such as manganese, 
phosphorus, etc., exert marked influences on the physical 
characteristics of steel. 

The modulus of elasticity of the steel wire used in 
the cables of long span, stiffened suspension bridges also 
has the value of about 30,000,000 pounds, the ultimate 
tensile resistance of such wire varying from about 200,000 
to 220,000 pounds per square inch. The resisting capacity 
of this material is largely affected by the process of cold 
drawing in its manufacture, but the modulus of elasticity 
seems to experience little or no effect of the cold working. 

Variation of Ultimate Resistance with Area of Cross-section. 

The ultimate resistance of a ductile material like steel 
depends to some extent upon the area of cross-section for 
a number of reasons. 

Generally the work put upon a bar of small cross-section 
in reducing between the rolls from the ingot or bloom to 
the finished bar will be greater for a bar of small section 
than for a similar bar of large section. Other things being 
equal, the greater amount of such work put upon the mate- 
rial the higher will be its physical qualities, including the 
ultimate resistance. Again, the temperature of a small bar 
or thin plate during its last passes between the rolls will 
generally be lower than for a bar of larger cross-section 
or for a thicker plate. In other words, the slight tendency 



Art. 58.] STEEL, 309 

toward cold rolling tends to enhanced ultimate resistance 
and elastic limit. 

Finally at the section of ultimate failure there is a 
'' necking down " to the final reduction of area of fracture 
within a short length of bar. This means a rather violent 
movement or flow of molecules of the material toward the 
axis of the bar, distinctly greater in distance for a larger 
bar than one of smaller section for the same percentage of 
final reduction. This corresponds to a greater longitudinal 
separation of the molecules near the axis of the specimen 
for a large bar than for a small one, which induces a little 
earlier rupture in the former bar than in the latter. 

For all these reasons the somewhat smaller ultimate 
resistance per square inch of cross-section is to be antici- 
pated for bars of large section, or plates of greater thickness 
than for bars of smaller sectional area, or for thin plates. 
This difference, how^ever, is much less for steel bars and 
plates at the present time than in the case of wrought iron 
when that material was widely or even exclusively used for 
structural purposes. 

Influence of Shortness of Specimen. 

If the dimensions of a test specimen are vSuch as to make 
exceedingly short that part within which failure will occur 
if a test is carried to rupture, there is less opportunity for 
the molecules of the material to move in toward the axis 
of the piece as failure is approached, thus preventing an 
unrestrained final reduction of fractured area. The result 
is an abnormal enhancement of the ultimate resistance. 
If the specimen is exceedingly short, as in the case of its 
being made by a groove, as shown in Fig. i, it is readily 
seen that the planes of shear indicated lie mostly in the 
enlarged part of the test piece. This condition prevents 
the free movement of the molecules along the oblique 



310 



TENSION. 



[Ch. VII. 



planes, required to produce the necking down or final 
reduction of area of section. In other words, the material 
at and in the vicinity of the section of failure is substan- 
tially supported by that in the enlarged part of the piece, 
thus enabling the ultimate section of fracture to retain an 
abnormally large area, which correspondingly raises the 
ultimate resistance. Many tests have been made with 
wrought-iron specimens to determine the limits of this 
influence of shortness. These tests show that the length 
of the reduced part of a test piece in which the section of 




fracture will be found should not be less than about four 
times the diameter in any case and that with ductile 
material five or six times would be preferable. As a matter 
of actual engineering practice, the length of the reduced 
part of a test piece is never less than about eight to ten 
times the diameter. In the case of a test piece of rect- 
angular section, the length should not be less than five 
or six times the greatest dimension of the cross-section, 
or preferably six to eight times that dimension. 

This matter of infiuence of shortness in test specimens 
is of the utmost importance in determining the true ultimate 
resistance of materials. If the test piece be too short the 
ultimate resistance will be unusually high. 



Elastic Limit, Resilience, and Ultimate Resistance. 

In scrutinizing the results of tests of specimens and 
full-size members of this section, it is to be observed that 



Art. 58.1 STEEL. 311 

the elastic limit is almost invariably the *' stretch-limit," 
or, as it is commonly called, * * the yield-point, ' ' and not the 
true * ' elastic limit, ' ' below which the ratio between in- 
tensity of stress and rate of strain is essentially constant. 
It has already been shown and stated that the true elastic 
limit is from 2000 to 3000 or 4000 pounds per square inch 
below the stretch-limit or yield-point. The stretch-limit is 
so readily observed without delaying the ordinary routine 
of testing that it has come to be called, although erro- 
neously, the elastic limit, in spite of the fact that it is a little 
above the intensity of stress to which that term should be 
applied. ' 

A 




Fig. 2. 

The elastic properties of three grades of steel are ex- 
hibited graphically in Fig. 2. The curved lines represent 
the tensile strains of the steel specimens at the intensities 
of stresses shown. The vertical ordinates are intensities 
of stress and the horizontal ordinates the rates of stretch, 
i.e., the stretches per unit of length, the latter being drawn 
20 times their actual amounts. The Rock Island Steel 



312 TENSION. [Ch. VII . 

belongs to a specimen of steel used for the combined rail- 
road and highway structure across the Mississippi River 
at Rock Island, 111., the data being taken from the U. S. 
Report of Tests of Metals -for 1896. The lines for axle 
steel and nickel steel are the graphical representations of 
data taken from the " U. S. Report of Tests of Metals" for 
1899. As in the previous case, the horizontal ordinates 
are the stretches per lineal inch shown at 20 times their 
actual values. The figures at the right-hand extremities 
of the curves are the ultimate resistances per square inch. 
The elastic limits and stretch-limits or yield-points are 
shown with clear definition. The remarkably hi^h elastic 
limit of the nickel steel is well indicated. 

By taking areas first between the horizontal axis OB 
and the inclined straight portion of each line, and then 
between the same horizontal axis and the entire line in 
each case, the following values of the elastic and ultimate 
resilience per cubic inch of each specimen will be found : 



Elastic Resilience 
Ultimate ' ' 



Rock Island 
Steel. 



24 in. lbs. 
10,500 in. lbs. 



Axle-steel. 



15.7 in. lbs. 
10,860 in. lbs. 



Nickel-steel. 



49 . 6 in. lbs. 
11,040 in. lbs. 



The three stress-strain lines or curves of Fig. i illus- 
trate completely the physical characteristics of the various 
grades of steel indicated under all degrees of stress up to 
actual failure, except that the lines are carried only to the 
maximum intensities of stress sustained. If those lines 
were prolonged to the actual parting of the metal, they 
would show rapidly descending portions like the broken 
portion of the Rock Island Bridge steel line. That por- 
tion of the curve, however, has little practical value, 
although considerable scientific interest. 

Table I contains a synopsis of the valuable series of 



Art. 58.] STEEL. 313 

tests of specimens by Prof. P. C. Ricketts. This table has 
already been explained on page 305. The tension tests 
show remarkabl}^ uniform results in elastic limit and ulti- 
mate resistance, and characterize a most excellent mate- 
rial. With the exception of the two Bessemer specimens 
containing 0.36 and 0.39 per cent carbon, all specimens 
were of mild steel. 

Table II exhibits results of tests of a number of un- 
usually large eye-bars 12 inches wide with other 8-inch 
and 7 -inch bars used in the Pennsylvania Railroad bridge 
across the Delaw^are River a short distance above Phila- 
delphia and completed in 1896. There will also be formd 
in the table tests of specimens taken from the same bars, 
together with the chemical composition. This table is 
interesting as disclosing the ultimate tensile resistance of 
large bars of mild steel having the chemical composition 
shown. The decrease in ultimate tensile resistance and 
elastic limit between the original bar and the finished eye- 
bar, due to the process of manufacture of the latter, is also 
evident at a glance. Although the steel in the original 
bars shows ultimate resistances revealed by the tests of 
specimens running from 58,300 pounds to 69,500 pounds 
per square inch, no ultimate resistance of the completed 
bars exceeds 59,500 pounds per square inch, while as small 
a value as 52,300 pounds per square inch is found. This 
table is taken from the description of the Delaware River 
bridge by Mr. F. C. Ktmz, Assistant to Vice-President 
of the American Bridge Company, Engineering Depart- 
ment, published at Vienna, 1901. 

Table III gives the results of testing a remarkable 
series of large steel eye-bars. The table exhibits not 
only the physical results of the tests but the chemical 
composition of the metal and the relative results for 
annealed and unannealed bars. Tl.e table was supplied 



314 



TENSION. 



[Ch. VII. 



Table II.* 
RESULTS OF TESTS OF EYE-BARS AND OF TEST SPECIMENS 







Full-size Eye-bars. . 






ii 




















3 


Pounds per Square 
Inch. 








1 1 s 















^ 




Length L, 


Length L2 


SS 






in Inches. 


in Inches. 


(X.O 








c 
















Cross-sec- 


c 


J' ^ '-n 










°C 




















tions xy 





^w ^ 
















i 


in Inches. 


Percent 
cess of 
over S 


Origi- 
nal. 


Final. 


Origi- 
nal. 


Final. 




Ultimate 
Strength. 


Elastic 
Limit. 


I 


12X2^ 


9 


37 


410 


472 


360 


414 


348 


55,000 


29,200 


2 


I2X2i 


9 


41 


407 


478 


360 


423 


434 


53,500 


25,900 


3 


I2X21I 


9 


41 


408 


471 


360 


416 


433 


53,750 


27,800 


4 


I2X2,V 


9 


42 


422 


487 


360 


418 


280 


58,300 


30,600 


S 


12X21^5 


9 


42 


420 


485 


360 


417 


284 


57,600 


30,000 


6 


12 X 21^5 


9 


42 


407 


483 


360 


430 


296 


56,300 


30,000 


7 


I2X1* 


9 


41 


403 


48s 


348 


423 


356 


53,000 


29,200 


8 


I2Xli 


9 


41 


400 


470 


348 


411 


332 


58,000 


31.400 


9 


I2X1H 


9 


38 


407 


471 


360 


419 


337 


54,800 


20,800 


lO 


I2X1I- 


9 


44 


407 


482 


360 


430 


256 


54,900 


29,300 


II 


12X11% 


9 


38 


400 


447 


348 


387 


414 


52,600 


29,200 


12 


12X11% 


9 


41 


415 


490 


360 


427 


266 


52,300 


29,900 


13 


I2XlT% 
12X11% 


9 


42 


411 


489 


360 


430 


293 


58,500 


30,600 






















IS 


8Xii 


8i 


47 


428 


483 


384 


434 


446 


53,500 


29,900 


i6 


8Xiii 


7i- 


34 


328 


394 


276 


334 


351 


53,300 


29,300 


17 


8Xiii 


6^ 


. 46 


328 


390 


276 


332 


199 


53,000 


29,300 


i8 


8Xi?B 


7^ 


40 


328 


388 


276 


328 


324 


58,500 


34,000 


19 


8X1 

7Xii 


6^ 


38 


470 


545 


432 


501 


306 


53,300 


32,900 








21 


7Xia 


8i 


44 


501 


575 


4S6 


525 


495 


59,500 


32,700 



, *From page 5, "The Delaware River Bridge, Built by Pencoyd Bridge Co.," by F. C. 
Kimz, "Allgem. Bauzeitung," Heft i, 1901. 




by Mr. Henry W. Hodge, C.E., of the firm of consulting 
engineers, Boiler & Hodge, of New York City. The varia- 
tion in chemical composition is accounted for by the fact 
that the bars tested were not specimens of any actual 
lot, but were forged and broken for the purpose of an 
investigation to determine specifications under which 
12 and 14 inch eye-bars for the ]\Ionongahela River canti- 
lever bridge should be manufactured. The machine in 
which the eye-bar heads were foiTned was not of sufficient 



Art. 58.] 



STEEL. 



315 



Table II. — Continued. 

TAKEN FROM THE SAME EYE-BARS, DELAWARE RIVER BRIDGE. 



Percentage of 


Test Specimen 8" Long, Approx. i" Square. 


i 

a 
"0 

i 







bod 
^ c 

ni 




(A 

(U 

m 


Chemical Composition 
in Per Cent. 


Pounds per 
Square Inch. 


Per Cent. 


No. 


C. 


S. 


P. 


Mn. 


Ulti- 
mate 
St'gth. 


Elastic 
Limit. 


Str'ch. 


Re- 
duc- 
tion. 


48.0 
36.0 


15.0 
17.7 
15.7 
16. 1 

15-9 
19.5 
21.5 

16.5 
19.4 
II . 2 
18.8 
195 


49-1 
44-7 
38.8 
43-2 
45.3 
38.4 
41.4 
49.3 
41 . 1 
41.2 
37.4 
54.3 
51.2 


0.18 


0.06 


. 04 


0.49 
0.62 


65,600 


33,200 


26.0 


45-4 


basic 

acid 

basic 

acid 
basic 

acid 
basic 


I 






















0. 03 










43 -o 
















36.0 


0.24 


0.05 


68,000 


40,300 


26. 75 


51-7 


5 
6 


44 -o 








8 

9 

10 

11 


49 -o 
44 -o 
42.0 

30.0 


0.24 
0.23 
0. 21 


0.08 
. 04 
0.07 

0.05 
0.07 
0.04 
0.05 
0.05 


0.06 
. 07 
0.06 


0.51 

0-55 
o- 51 


69,500 
66,700 
63,800 


36,200 
36,300 
35,800 


25.25 
27.25 
27 .00 


42.9 

55.2 
39.5 


45 -o 
47 -o 


0.24 
0.17 
0.21 
0. 20 
0. 20 


0.0s 
0.06 
0.02 
0.03 
0.03 


0.65 
0.58 
0.45 
0.56 
0.56 


66,500 
65,000 
59,500 
60,300 
58,300 


39,000 
36,800 
33,700 
32,000 
30,800 


28.50 
26.00 
33-50 
28.00 
29.25 


47.9 

57.3 
45.6 
58.9 
52.7 


13 
14 

\l 

17 
18 


3Q-0 
40.0 
33-0 


13. 1 
21 . 2 
20.5 
19.7 
16. 1 


44.9 
41.9 

29.7 
48.6 
49-3 


42.0 












42.0 


0. 22 


0.08 


0.06 


0.59 


62,400 


35,600 


30.2s 


58.0 


20 
21 




15.4 


45-8 





















capacity to give satisfactory results, and hence it will be 
observed that most of the bars broke in the head or neck. 
The actual bars for the structure were to be forged in a 
new machine of greater capacity. 

The results are highly interesting as indicating what 
excellent results may be obtained even for the largest bars 
under satisfactory conditions of manufacture. 

Shape Steel and Plates. 

A number of specimen tests were made, during 1899- 
190 1, of the open-hearth acid and basic steel shapes and 
plates for the construction of the City Island bridge and 
the 145th Street bridge across the Harlem River, both 
in the city of New York, under the direction and super- 



3i6 



TENSION. 



[Ch. VIL 



Table III. 
REvSULTS OF FULL-SIZE EYE-BAR TESTS ON TRIAL STEEL. 
MONONGAHELA RIVER CANTILEVER. BOLLER & HODGE, 
CONSULTING ENGINEERS. 
The steel was basic open-hearth metal manufactured and rolled by the 
Carnegie Steel Company, 1902. All bars were about 30 feet long. 
"A" means annealed and "N" not annealed. 



Specimen. 





Chemical Composition. 


Bar. 












Car. 


Phos. 


Mang. 


Sulph. 


12" Xlf" 


.30 


.021 


.62 


.026 \ 


12" Xlf" 


.28 


.020 


.54 


.022 ■ 


12" Xi|" 


.26 


.03 


.52 


.0. 1 


12" Xrf" 


.36 


.019 


.5; 


.035 ] 


12" Xlf" 


.32 


.03 


.51 


•03 I 


12" Xlf" 


.28 


.03 


.46 


.04 1 



Elas. 


Ult. 


Elonga- 
tion, 






Percent 


Pounds per Sq. In. 


in 8" . 


A 37380 


67600 


27.0 


N 44330 


71080 


21.5 


A 32050 


60230 


26.5 


N 37230 


68000 


25-7 


A 38610 


69700 


28.7 


N 39700 


72400 


27.2 


A 31250 


70720 


28.5 


N 37740 


76980 


18.7 


A 29140 


67120 


28.5 


N 35600 


72120 


25.0 


A 37050 


69820 


30.5 


N 40450 


73340 


27.5 



Redact. 





Full-size Bar. 




Bar. 


Elas. 


Ult. 


Elongation, 
Per cent, 
in 20 Feet. 


Reduct., 
Per cent. 


Remarks. 




Pounds per Sq. In. 




12" X if" 
12" Xlf" 
12" Xlf" 
12" Xlf" 


38190 
37480 
33330 
34320 
34210 
35930 


64140 

58670 

61090 , 

63030 

59170 

65520 


15-05 
5-7 

11-55 
7 -05 
6.0 
9-56 


55 - 54 


Broke in body. 
" " neck. 

" " head. 

(( <( (( 






12" Xlf" 
12" Xlf" 


5.96 


" " bodv. 
" " head. 





vision of the author. Table IV contains the results of 
a portion of such tests for the quaHty of material used. 
It will be seen that the specimens were taken from a 
wide range of shapes and plates, and that a large portion 
of the material was produced by the basic open-hearth 
process. The table is of special value in consequence 







gk 






• 




i 


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15x2 

PWENIX IRON CO. 


\ 


"^'^Vl^F 




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I 


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s 



[5 X 2-in. steel eye-bar forged at the shops of the Phoenix Bridge Co., Phoenixville, Pa. 
The bar developed an ultimate resistance of 50,160 lbs: per sq. in. and 28,000 lbs. per 
sq. in. at elastic limit. The elongation in 8 ins. of the bar, including the section of 
failure, was 25.6 per cent, and the elongation of the pin-hole was 5.26 inches. The 
reduction of area at the section of fracture was 52.9 per cent. 

(To face page 316.) 



Art. 58. 



STEEL. 



317 







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3i8 TENSION. [Ch. VII. 

of the wide range of sections covered by it, as well as for 
the chemical data which it contains, showing the percent- 
ages of carbon, manganese, phosphorus, and sulphur 
contained by the steeL Both the chemical analyses 
and the physical results indicate that many of the shapes 
are of mild steel, while the remaining portion is of low 
steel. 

The quality of metal either in steel shapes or plates 
depends largely upon the amoimt of reduction reached 
in the passage of the blooms through the rolls before the 
final area of section is attained. In the early days of 
rolling steel sufficient work between the rolls was not 
always done, and the quality of the metal suffered corre- 
spondingly. This defect is seldom or never found at the 
present time and the corresponding variations in certain 
physical qualities are avoided. In the case of wide and 
thin plates, in which the temperature of the metal may 
be lower than in thicker plates at the last pass through 
the rolls, increased hardness may sometimes be found, 
but as a rule there will be little, if any, difference, as the 
preceding tables show, in the physical results for the thick 
and thin sections ordinarily used in engineering construction. 

Tests of specimens from a large variety of shapes, 
plates, and bars used in the towers and stiffening trusses 
of the Manhattan Suspension Bridge across the East River 
at New York City, as given in the Report of Mr. Ralph 
Modjeski, consulting engineer, 1909, show the following 
results : 

Carbon Steel for Towers. 

Metal from plates, bars, channels, bulb angles, 
and rivet rounds gave average elastic limits for 
different sets of tests varying from a maximum of 
43,040 pounds per square inch down to 31,137 



Art. 58.] STEEL. 319 

pounds per square inch. The average ultimate 
resistances of the same sets of tests gave a maxi- 
mum of 65,880 pounds per square inch with inter- 
mediate values running down to 51,380 pounds 
per square inch. The smaller of each of these sets 
of results belongs to the low steel used for rivets; 
the higher values belong to shapes, plates and bars. 

Carbon Steel for Suspended Structures. 

Tests of specimens cut from shapes, plates, bars 
and rivet rounds gave average elastic limits run- 
ning from a maximum of 44,505 pounds per square 
inch down to 33,907 pounds per square inch. The 
corresponding ultimate resistances varied from 
68,652 pounds per square inch down to 52,411 
pounds per square inch. Again, the smaller values 
are found for the low-carbon rivet steel. 

Nickel Steel for Stiffening Trusses. 

Tests of specimens cut from nickel-steel shapes, 
bars and rivet rounds used in the suspended 
structure gave average elastic limits for different 
sets of tests varying from a maximum of 61,355 
pounds per square inch down to a minimum of 
55,400 pounds. The corresponding ultimate resist- 
ances varied from a maximum of 90,760 pounds 
per square inch down to 77,268 pounds per square 
inch. 

The preceding results for the Manhattan Suspension 
Bridge show values which may reasonably be expected for 
such carbon and nickel steels as are now in use for the best 
types of large bridge structures. The carbon steel for the 
plates, shapes and bars belongs to the grade of medium 



320 



TENSION. 



[Ch. VII. 



steel with ultimate tensile resistance running from about 
60,000 pounds per square inch to 68,000 pounds per square 
inch. The carbon content varies approximately from .2 
to .30. The nickel content in the nickel steel was 3.25 
per cent. 

Steel Wire. 

The process of production of steel wire is one of cold 
drawing which, like all cold work, increases ultimate 
resistance and elastic limit, but decreases ductility. 

The physical properties of wire will depend to a con- 
siderable extent upon its diameter; the smaller the latter, 
as a rule, the greater will be the ultimate resistance. 

TENSILE TEST OF SA^IPLES OF WIRE. 
Diameter of Wire=o.iq8 Inch. 



No. of Test 



Av. 



Unit stress at first 
measurable set 
(wire as received 
lb. per sq. in.) . . . 

Unit stress at first 
measurable set 
(second applica- 
tion of stress lb. 
per sq. in.) 

Unit stress at Limit 
of proportional- 
ity 

(wire as received) . . 

Unit stress at yield 
point lb. per sq. 
in. 

Ult. unit stress lb. 



45,000 



72,500 



46,000 



43,500 

119,500 
83,500* 



per sq. m. 

Elongation in 8 in. 

per cent 

Reduction of area 

at fracture, per 

cent 



136,500 
212,300 212,200 



131,500 
215,200 



31.4 27.2 



23.0 



109,000 



i07,ooot 



210,400 



350 



24.7 



2 i 2,000 



463 



25.2 



44,830 



119,500 



78,000* 
io9,ooot 



134,000 

212,400 

4.065 

26.30 



* Elongation measured over 8 ins. gage length with Ewing Extensometer reading to 
one-fifty-thousandth inch. 

t Elongation measured over 2.5 ins. gage length with Ames Test Gauge reading to one 
ten-thousandth inch. 



Art. 58.] STEEL WIRE AND CASTINGS. 321 

The quality of carbon-steel wire for the latest type of 
large stiffened suspension bridge is well exemplified by that 
used in the construction of the cables of the Manhattan 
Suspension Bridge also built across the East River at New 
York City with a span 1470 feet between centres of towers, 
each of the tv/o side spans being 725 feet from centre of 
tower to centre of shore end support. This wire, about 
|-inch in diameter, was tested b}^ Mr. Ralph Modjeska and 
his report of 1909 gives the above results. (See Table at 
bottom of p. 320.) 

The specifications prescribed the following chemical 
requirements for this cabbie wire : 

Carbon, not to exceed 85 of one per cent. 

Manganese, not to exceed 55 of one per cent. 

Silicon, not to exceed 20 of one per cent. 

Phosphorus, not to exceed 04 of one per cent. 

Sulphur, not to exceed .04 of one per cent. 

Copper, not to exceed 02 of one per cent. 

Some special wires of small diameter give exceedingly 
high resisting capacity. 

In the U. S. Report of Tests of Metals for 1897 will 
be found the results obtained by testing piano wire of 
different diamxCters, as follows : 



Wire. 


Ultimate Resistance, 
Pounds per 
Square Inch. 


Final Contraction. 


No. 14 (.033 in. Diam.) 
No. 23 (.0513 in. Diam.) 


357,440 to 389,470 
325,110 to 337.200 


28 to 37.9% 
42.2 to 45.1% 



Steel Castings. 

The results exhibited in Table V belong to the 
steel castings used in the turntable of the draw-span of 



322 



TENSION. 



[Ch. VII. 



the 145th Street bridge across the Harlem River in New 
York City. The tests were made in 1901. The left- 
hand column of the table shows the particular (cast) 
members of the turntable from which the specimens 
were taken. They also show that, in steel castings, a 
sensibly higher grade (in the sense of containing more 
carbon and manganese) of steel is used than in rolled 
shapes. As indicated in the heading of the table, the 
material was acid open-hearth steel. The ultimate tensile 
resistance runs from about 67,000 pounds to nearly 76,000 
pounds per square inch. The elastic limit is also obser\^ed 
to be high, in consequence of the rather large percentage 
of manganese. The quality of metal exhibited by the 
physical results of the table is fairly representative of that 
ordinarily used in steel castings. Obviously the ductility 
exhibited is less than that foimd in connection with rolled 
shapes. 

Table V. 
TENSILE TESTS OF ACID OPEN-HEARTH STEEL CASTINGS, 1901. 



specimen from 



Loads in Poiinds 
per Sq. In. 



Elastic 
Limit. 



Ulti- 
mate. 



c ^ 


i^ 


Chemical Analysis. 


..-. >—t 


c 






_ c 

5"u 














s^ 




c. 


Mn. 


P. 


s. 


Si. 


31.2 


39.8 


•23 


.70 


.0521.004 


.27 


30-4 


29 




.27 


.65 


.045-018 


.26 


34-3 


46 


5 


.27 


.60 


.047 .007 


.28 


32.0 


40 


8 


•30 


.65 


.040 .004 


.28 


31.2 


39 


8 


■23 


.70 


.052 .004 


.27 


32.6 


44 


5 


•27 


.60 


. 047 ^ . 008 


.28 


29.6 


38 


5 


•23 


.60 


.052,. 004 


.27 


30.4 


42 


5 


•30 


.65 


.041 .009 


.26 


28.1 


37 


6 


.29 


.70 


. 046 . 006 


• 25 


31-4 


39 


4 


.29 


.70 


.046 .006 


• 25 


21.9 


37 


I 


.29 


.70 


. 046 . 006 


• 25 


31.6 


31 


7 


■30 


.65 


•0371.004 


.27 


29.6 


37 


7 


■25 


•50 


.043 .02 


.20 


31.2 


40 




.29 


•65 


.05 .024 


.27 


29.6 


45 


5 


•31 


.60 


.036 .003 

1 


.26 



Character 

of 
Fracttire. 



Turntable wheel. 
Track segments. . 



Rack segments . 
Track segments 



Turntable v/heel 

Rack segments. . 
Track segments. 
Shoes 



47,510 
49, 500 
47,500 
47,500 

49,775 
47,500 
46,270 
48,900 
46,130 
48,640 

49,77 5 
47,600 
45,200 
47,500 
49,800 



72,300 
67,200 
67,900 
71,100 

72,115 
68,100 
71,920 
74,700 
71,860 
71,600 

71,335 
76,020 
68,000 
68,700 
75,700 



Silky cup. 

" ang. 
Irregular. 
Silky ang. 

" cup. 
Irregular. 
Silky cup. 



ang. 
Irregular. 



Art. 58.] 



STEEL. 



323 



Rail Steel. 

The grade of steel adapted to railroad rails is much 
higher in the hardeners carbon and manganese, and corre- 
spondingly higher in physical quantities than structural 
steel, at the same time it is a quite different metal from 
that adapted to the finer purpOvSes of tools ; it is manufac- 
tured by the Bessemer process. The great increase in 
the immediate past in the weight and speed of railroad 
locomotives and trains has subjected rails to intensely 
severe duties which can be performed without deteriora- 
tion of metal only by steel of the highest powers of en- 
durance, which means a steel of high ultimate resistance, 
elastic limit, and corresponding ductility. The grades 
of steel used for rail purposes at the present time are 
well illustrated by the following tabular statement, which 
shows the chemical composition of the rails of various 
weights and sizes used by theN. Y. C. & H. R. R. R. Co., 
the pounds at the head of the columns indicating the weight 



NEW YORK CENTRAL & HUDSON RIVER R. R. SPECIFICATIONS. 



65-Lb. 70-Lb 



7 5 -Lb. 



5o-Lb. 



loo-Lb. 



Carbon. 



Silicon. 



Manganese. 



Sulphur not to exceed 

Phosphorus not to exceed 

Rails having carbon below will be 

rejected 

Rails having carbon above will be 

rejected 



0.45 
to 

0.55 
0.15 

to 
0.20 
1.05 

to 

1-25 

0.069 
0.06 

0.43 
0.57 



0.47 

to 
0.57 
0.15 

to 
0.20 

1.05 

to 

1.25 

0.069 
0.06 

0.45 
0.59 



0.50 

to 
0.60 
0-15 

to 
0.20 
1 . 10 

to 

1.30 

0.069 
0.06 

0.48 

0.62 



0.55 

to 
0.60 

0.15 

to 
0.20 
1 . 10 

to 

1.30 

0.069 
0.06 

0.53 
0.65 



0.65 

to 
0.70 

015 

to 
0.20 
1 .20 

to 
1 .40 
0.069 
0.06 

0.60 

0.70 



The numbers represent the per cents of the various elements. 



324 TENSION. [Ch. VII. 

of rail per yard. The metal of the lightest or 6 5 -pound 
rail corresponds to an ultimate resistance of 85,000 to 90,000 
pounds per square inch, with an elastic limit of .5 to .7 of 
that value. The highest or 100-pound rail corresponds to 
metal having an ultimate tensile resistance of probably 
110,000 to 120,000 X->ounds per square inch, with an elas- 
tic limit of .6 to .7 of those amounts. In these chemical 
compositions it is pertinent to observe the high carbon 
and manganese, and the low phosphorus and sulphtir. 

After several years' experience in the effort to secure 
a most enduring steel for a railroad rail weighing 135 
pounds per yard, Mr. James 0. Osgood, Chief Engineer 
of the Central Railroad of New Jersey, states in a paper 
published in the Official proceedings of the New York Rail- 
road Club for May 21, 191 5, that the following chemical 
composition has ^delded the most satisfactory results within 
the experience of that road, on which, where these heavy 
rails are laid, the traffic is of excessive intensity. 

Carbon .85 to i.oo per cent or carbon .8 to .95 per cent. 

The rails having the latter carbon content also contain 
chromium^ 0.2 to 0.4 per cent and nickel 0.2 to 0.4 per cent. 
It Vv'ill be observed that this rail section, i.e., 135 pounds 
f)er yard, is the heaviest yet rolled and used in the United 
States up to the date of Mr. Osgood's paper. 

Rivet Steel. 

The grade of steel ordinarily used for rivets is the 
softest, or lowest in hardeners, employed in engineering 
construction; it should thus be correspondingly low in 
phosphorus and carbon. In Table I of this article 
there Avill be found the measures of ductility and other 
physical properties of a number of specim.ens of rivet 
steel, which are fairly representative of that metal, except 



Art. 58.] STEEL. S^S 

tlmt the ultimate resistance is frequently much lower 
than is shown there. In much of the rivet metal used 
at the present time the ultimate tensile resistance may 
run from 52,000 to 60,000 pounds per square inch. In 
such steel the carhon may n,ui down to .06 or .08 per cent. 
with sulphur between .02 and .03 per cent., and phos- 
phorus even lower. The treatment to which rivet metal 
must be subjected in the heading of rivets makes it 
imperative that the metal possess qualities of ductility 
and toughness to an unusual degree and that the vari- 
ations of temperature in the rivet shall not reduce its 
resisting capacity. In other words, rivet steel must pos- 
sess physical properties enabling it to resist torturing 
treatment to the highest practicable degree. 



Nickel Steel, 

The alloy, nickel steel, to which the allusion has already 
been made in connection Vv'ith the subject of the modulus 
of elasticity of steel, possesses marked characteristics of 
high ultimiate resistance and elastic limit, the latter usually 
running from j% to f of the former. The amount of nickel 
in the alloy is usually about 3. 2 5 -per cent, while the carbon 
content may frequently be .25 to .30 per cent, although 
higher values of the nickel content will be found in the table 
following, which shows the results of tests of both full-size 
eye-bars and specimens cut from those bars. That table* 
shows the high ultimate resistance and elastic limit ^delded 
by this material, with but little if any decrease in ductility. 
The effects of annealing may be observed to be practically 
the same as for carbon steel. 

* The results in this table were courteously given to the author by Mr. 
HenryW. Hodge, C. E., of the firm of consulting engineers, who designed and 
built the St. Louis Municipal Bridge, at St. Louis, Mo. 



326 



TENSION. 



[Ch. VII. 



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Art. 58.] 



STEEL. 



327 






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3^8 



TENSION. 



[Ch. VII. 



The following tabular statements give the physical 
qualities of nickel steel adapted to the various purposes 
indicated. They are taken from results published in the 
Railroad Gazette for August 8th, 1902. 

NICKEL-STEEL FORGINGS. 



Tensile 
Strength.Lbs, 



Elastic 
Limit, Lbs. 



Exten., 
Per Cent. 



Cont., 
Per Cent 



Driving-wheel axles 

Piston-rods 

Main crank-pins 

Front crank-pins 

Connecting-rods and guides 



99,310 
90,140 
93,570 
92,180 
92,040 



64,170 
60,090 
65,450 
64,170 
59,820 



25.00 
25.50 
24.00 
24-50 
26.00 



NICKEL-STEEL CASTINGS. 



Crosshead 

Furnace-bearer, bearer-guide 
Annealed : 

Carbon steel 

Nickel steel 

Oil-tempered : 

Carbon steel 

Nickel steel 



84,540 
85,050 

109,500 
100,330 

129,360 
103,890 



53,980 
54,490 

51,440 
66,720 

67,230 
76,390 



18 


1 
50 


18 


00 


19 


50 


25 


00 


17 


50 


25 


00 



31- 10 
26.04 

36.31 
54.56 

38.53 
61.56 



SMALL RIFLE BARRELS— NICKEL STEEL. 



Tensile Strength, Lbs. 


Elastic Limit, Lbs. 


Ext. in 2 Inches, 
Per Cent. 


Cont. or Area 
Percent. * 


115,100 
114,080 
114,590 
116,620 
116,120 
114,590 


99,820 
97,780 
99,820 
96,770 
97,780 
98,800 


23 
23 
23 
22.50 

23 
24 


64.00 
64.95 
65.45 
62.05 
64.00 
62.53 



Vanadmm Steel. 
The alloy called vanadium steel contains when used for 
many purposes some chromium, which frequently gives it 
the name Chrome Vanadium Steel. This grade of steel 
contains carbon and manganese about in the proportion of 
ordinary structural steel. Indeed it may be considered 



Art. 58.] 



STEEL. 



32( 



ordinary structural steel alloyed with chromium and vana- 
dium. The addition of these latter materials gives to the 
resulting product great toughness with high ultimate resist- 
ance and an elastic limit remarkably high in proportion 
to the ultimate resistance. It is used largely for such 
special purposes as locomotive parts, both as castings and 
in the forged condition. In either case, however, it requires 
heat treatment. It is largely used for locomotive frames, 
axles, piston rods, crank pins, tires, as well as for many 
parts of automobiles. 

Many physical tests of small specimens have been made 
giving elastic limits of about 40,000 pounds per square inch 
(for castings) up to about 100,000 pounds per square inch, 
the corresponding ultimate tensile resistance being about 
70,000 pounds per square inch up to about 150,000 pounds 
per square inch. These variations in physical qualities 
depend upon chemical contents of the alloy and upon the 
condition of the material as cast or rolled, and finally upon 
the heat treatment of the material. 

In a paper on '' Vanadium Steel in Locomotive Con- 
struction " by George L. Norris, Engineer of Tests of the 
American Vanadium Co., published in the Official Proceed- 
ings of the New York Railroad Club, 191 5, he gives the 
following chemical contents as meeting the requirements 
for the locomotive parts indicated. 

Chemical Contents of Chrome Vanadmm Steel. 





Castings 


Axles, Piston Rods 
and Crank Pins 


Tires 


Carbon 


.20 to .30% 


.30 to .40%* 


.50 to .65% 


Manganese 


.50 to .70 


.40 to .60 


.60 to .80 


Chromium 





•75 to 1.25 


.80 to 1. 10 


Silicon 


.20 to .30 


Not over .20 


.20 to .35 


Vanadium 


Over .16 


Over .16 


Over .16 


Phosphorus 


Not over .05 


Not over .04 


Not over .05 


Sulphur 


Not over .05 


Not over .04 


Not over .05 



* Preferred .35% 



330 



TENSION. 



[Ch. VII. 



The elastic limit, ultimate resistance, final stretch and 
final reduction of area corresponding to the grades of mate- 
rial indicated by the chemical contents are shown in the 
next table. 



PHYSICAL REQUIREMENTS (After Heat Treatment), 





Elastic Limit 


Ult. Resist. 


Stretch in 


Reduction 




Lbs. per sq. in. 


Lbs. per sq. in. 


2 ins. 


of Area. 


Castings 


40,000- 50,000 


70,000- 85,000 


25% 


45% 


Axles, Piston 










Rods and 










Crank Pins. . . 


80,000-100,000 


95,000-125,000 


25 


55 


Tires 56" diam. 










and under .... 


110,000-125,000 


140,000-160,000 


Min. 12 


Min. 30 


Tires over 56'' 










diam 


95,000-115,000 


120,000-140,000 


Min. 15 


Min. 35 



These physical requirements correspond closely to the 
usual results of tests. They show the high elastic limit of 
the material and its high degree of ductility. 

Castings must be carefully annealed by heating slowly 
to about 1550"^ F. and then slowly cooling. 

The heat treatment for chrome vanadium driving axles 
consists of: 

" (i) annealing the rough forging by heating carefully 
and cooling slowly, (2) reheating,"forging, and quenching in 
water or oil, preferably the latter, (3) then promptly re- 
heating slowly and uniformly to a temperature sufficiently 
high to give the desired properties. The forging must be 
held at this final or draw-back temperature for at least two, 
hours. The axle should then be allowed to cool slowly. 

" The recommended temperature for annealing is 1475- 
1525° F., and for quenching from 1600° F. to 1650° F. 
The final heating for obtaining the physical properties 
should be approximately 1100° F. to 1200° F." 

The heat treatment to which vanadium side rods, piston 



Art. 58.] STEEL. 331 

rods, and crank pins are submitted is the same as that 
given above for driving axles. 

In the manufacture of locomotive tires, the heat treat- 
ment is somewhat different from that set forth above, as 
it consists of : 

'' (i) In reheating the tires after rolling, and then 
quenching in oil, (2) then reheating slowly and uniformly 
to a temperature sufficient!}^ high to obtain the desired 
physical properties. The tire must be held at this final 
temperature at least two hours, which is considered the 
minimum time required for the changes to be effected 
throughout the tire section. The tire should then be with- 
drawn from the furnace and allowed to cool in still air. 

*' The recommended temperature for quenching is about 
1600° F. The final heating for obtaining the physical 
properties specified should be approximately iioo to 
1200 h. 

It is obvious that material with such physical properties 
possesses unusual toughness and resilience. For that reason 
it is specially adapted to locomotive springs and other 
similar uses. For such a purpose the carbon contained is 
relatively high. Mr. Norris in the paper already indi- 
cated gives the following as a suitable chemical composition : 

Chemical Composition. 

Per cent. 

Carbon 0.52 to 0.60 

Manganese 0.70 to 0.90 

Chromium 0.80 to ii.o 

Vanadium Over o. t6 

Phosphorus . .' Not over 0.04 

Sulphur Not over 0.04 

This material requires heat treatment consisting of : 

" (i) Heating and quenching in oil, (2) then reheating 



332 



TENSION. 



[Ch. VII. 



or drawing back, preferably in a lead bath, and cooling 
slowly. The time in the lead bath should be lo to 15 
minutes. 

"The recommended temperature for quenching is from 
1575 to 1650° F. The drawback or annealing temperature 
should be approxim.ately from 900 to 1100° F." 

When such material is tempered for railway springs it 
has the following physical properties : 

" Elastic limit, lbs. per sq.in 160,000-180,000 

Tensile ^strength, lbs. per sq.in. . . 190,000-230,000 

Elongation in 2 inches 10-15% 

Reduction of area. 30-45% " 

This material possesses the highest physical properties 
of the steels yet used for commxCrcial purposes. 

Some recent tests, June, 191 5, reported by the American 
Vanadium Company, show excellent results for carbon- 
vanadium steel both in the natural condition of the vSpeci- 
mens and after simple annealing as well as after heat treat- 
ment, the latter yielding highest results generally, but not 
for ultimate resistance, the ductility, however, being dis- 
tinctly lower in the natural condition. The following table 
gives the results of the tests as well as the chemical analysis 
and treatment. The first six- sets of values belong to test 
specimens taken from 7 -inch and 11 -inch axles, w^hile the 
last three belong to specimens from connecting rods. 

TESTS OF CARBON-VANADIUM STEEL 





Carbon 


Manganese 


Phosphorus 


Sulphur 


Vanadium 


Chem. Analysis. . 


0.47% • 


0.90% 


0.012% 


0.020% 


0.15% 



Art. 58.] EFFECT OF HIGH AND LOW TEMPERATURES. 333 

PHYSICAL PROPERTIES 



Treatment. 



Yield 


Elastic 


Ultimate 


Stretch 


Point 


Limit 


Resist. 


m 2 in. 


lbs. per 


lbs. per 


lbs. per 


Per- 


sq. m. 


sq. m. 


sq. m. 


cent. 


71,200 


68,000 


123,000 


16.0 


56,000 


52,000 


90,000 


24.0 


85,000 


82,000 


112,500 


22.0 


75,000 


70,000 


117,000 


16.0 


58,000 


54,000 


94,000 


22.0 


87,000 


80,000 


115,000 


20.5 


92,000 


85,000 


131,000 


17.0 


71.500 


67,000 


105,000 


235 


92,500 


86,000 


123,000 


20.5 



Reduc- 
tion of 
Area 
Per- 
cent. 



Natural 

Annealed 1450° F 

O. Q. 1600; T. 1160° F 

Natural 

Annealed 1450° F 

O. Q. 1600; T. 1160° F 

Natural 

xA.nnealed 1450° F 

O. Q. 1600; T. 1160° F 



30.0 
50.0 
550 

28.5 
47 
52.0 

44.0 
52.0 
50.0 



Effect of Low and High Temperatures on Steel. 

There has been much difference of opinion expressed 
upon the effect of low temperature upon steel, especially 
upon steel rails. The high number of breakages in steel 
rails during the winter, particularly in the early days of the 
use of steel for such a purpose, has given the impression 
that low temperatures in the vicinity of zero degrees F., 
or lower, make steel brittle and hence subject to sudden 
fracture without warning in the cold weather of winter. 
This impression has been shown to be without material 
foundation in rails of the best quality, but phosphorus 
makes iron and steel " cold short." If, therefore, there 
should be a sufficient amount of phosphorus present steel 
or iron would undoubtedly become more liable to fracture 
at low tem.peratures. In the early days of rail making, 
when the constituent elements were not so carefully con- 
trolled as at present, it is highly probable if not practically 
certain that the presence of phosphorus accounted for many 
breakages at low temperatures. For many years, however, 



334 TENSION. [Ch. VII. 

the effects of the prejudicial hardeners phosphorus and 
sulphur have been well recognized and they have been kept 
so low as to have no material effect upon the finished 
products. 

Again, frozen ground in the winter adds somewhat to 
the rigidity of a roadbed, enhancing to some extent the 
effects of shocks or blows to which rails are subjected under 
rapidly moving heavy train loads. Some of the increased 
breakages in the winter are probably due to this cause and 
it is possible that a great majority of them may be ac- 
counted for in this way. 

On the whole the latest experiences do not seem to 
indicate that with the excellent quality of steel now pro- 
duced for engineering purposes the effects of low tempera- 
tures are at all serious, but that they may be ignored when 
suitable precautions are taken in the processes of manu- 
facture. 

The effect of high temperature, on the other hand, is a 
matter of some concern in connection with building con- 
struction, since the ultimate carrying capacity of iron or 
steel may be seriously affected or even destroyed by the 
high temperatures of conflagrations unless the supporting 
members are protected against the effects of intense 
heat. 

Figs. 3 and 4 represent the results of investigations by 
Prof. R. C. Carpenter, formerly of Cornell University, who 
made tensile tests on wrought iron and steel circular speci- 
mens .5 inch in diameter. Fig. 3 is self-explanatory. It ' 
shows the graphical relation between the temperatures of 
the specimens and the ultimate tensile resistance per ^square 
inch. 

The ductility represented by the final elongations 
or stretches in 8 inches at the corresponding temperatures 
of rupture are exhibited in Fig. 4. 



Art. 58.] 



STEEL. 



335 



Prof. Carpenter observes " that all the curves have 
a point of contrafiexure at about 70° F., and another 
at a temperature not far from 500°. The maximum 
strength is found at temperatures c-f zi.oo° to 550°. At 



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CIMEN, DEGREES F. 



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temperatures higher than this all the materials show a 
rapidly decreasing strength." 

As a general result or consensus of all results, includ- 
ing the older and the later, it may be stated that iron 
and steel lose no sensible portion of their resisting capac- 
ity under about 500° Fahr., but that softening is liable 
to begin when the temperature rises much above that 
limit. At a temperature of about 800° Fahr. these metals 
may lose as much as 20 per cent, of ultimate resistance. 



33^ 



TENSION. 



[Ch. VII. 



Hardening and Tempering. 
The processes of hardening and tempering are not 
usually applied to structural steel, but to those higher 
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TEMPERATURE OF SPECIMEN, DEGREES F. 



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900 



Fig. 4 
or wire. The hardening process consists in heating the 
steel to such temperature as may be desired to accomplish 
a given purpose and then quenching in water, brine, 
oil, molten lead, or other proper bath. The temperature 
from which the quenching is done m.ay be that indicated 



Art. 58.] STEEL. 337 

by an orange color; it depends upon the size or character 
of grain of metal desired. In general terms, the higher 
the content of carbon, the more marked will be the re- 
sults of the hardening processes. Quenching has a com- 
paratively small effect upon low or medium structural 
feteel. 

Tlie process of tempering is, in reality, supplementary 
to the process of hardening in the manner just described. 
After a piece of steel has been hardened by quenching 
so that its temperature is that of the air, if it be again 
heated it will exhibit different colors as the temperature 
is increased. The first noticeable color will be a light 
delicate straw, then deep straw, light brown, dark brown, 
brownish blue, called "pigeon wing," light bluish, light 
brilliant blue, dark blue, and black, after which the temper 
is completely remioved. The preceding colors are due 
to thin films of oxide that form on the exterior surfaces 
of the pieces as the temperature increases. When this 
heating is stopped at any color and the steel allowed to 
cool, the metal is said to be drawn to the temper shown 
by the corresponding color. 

The tempers at different colors for different processes 
are sometimes stated as follows: 

Light straw For lathe-tools, files, etc. 

Straw " " " " " 

Light brown " taps, reamers, drills, etc. 

Darker brown " " " " " 

Pigeon wing " axes, hatchets, and some tools. 

Light blue " springs. 

Dark blue " some springs, occasionally. 

Tempering or hardening increases both the elastic 
limit and ultimate resistance, but decreases the ductility. 



33^ TENSION. [Ch. VII. 

Annealing. 

The processes of annealing, like those of hardening 
and tempering, produce more marked results in the higher 
steels than in the lower. Steel has a sensibly varying 
density at different temperatures; in other words, a 
given weight of metal will occupy sensibly different volumes 
at different temperatures. Hence if a piece of steel be 
subjected to any operation, such as forging, which gives 
to different portions concurrently widely varying tem- 
peratures, those portions will necessarily be subjected 
to considerable intensities of internal stresses, and if 
those stresses are not rem^oved they may reduce greatly 
both the ultimate resistance and ductility. In the higher 
grades of steel and in special steels it is, therefore, impera- 
tive to anneal members which have been subjected to 
such operations. These observations are specially perti- 
nent to such high steels as those adapted to the manufacture 
of tools or other similar purposes. In general it is neces- 
sary in structural engineering practice to resort to anneal- 
ing only in the case of eye -bars, or other members which 
have been subjected to the operations of forging. The 
process consists simply in heating the member to be 
annealed to about a cherry-red temperature until the 
piece is heated through, and then allowing it to cool grad- 
ually to a normal temperature. At the cherry-red heat 
the metal is sufficiently softened to allow the molecules 
to readjust their relative positions so as to remove the 
internal stresses. After the operation of cooling is com- 
pleted the metal will be at least approximately, if not 
entirely, in a condition of no internal stresses, i.e., if the 
annealing has been properly done. The more gradually 
and uniformly the cooling is accomplished the more ex- 
cellent will be the results. Sometimes resort is made to 



Art. 58.] STEEL. 339 

such specia.l means to accomplish these ends as covering 
the members, after bringing them to a proper temperature, 
with sand, ashes, or other similar material, to insure a 
slow and uniform cooling. 

The preceding tables show what is alwa^^s found in 
a comparison of results for the natural and the an- 
nealed metal. The process of annealing will diminish 
the ultimate resistance of structural steel in general from 
about 4,000 to 6,000 or 8,000 pounds per square inch, 
and the elastic limit will be reduced correspondingly. 
These effects will be found more marked as the metal is 
finished between the rolls at lower temperatures. In 
general, steel which is hardened by the conditions of 
manufacture, like that of comparatively low temperature 
in rolling, will exhibit greater decreases of ultimate resist- 
ance and elastic limit under annealing. 

The process of annealing increases the ductility of 
the steel, since it softens the metal. In spite of the re- 
duction in ultimate resistance and elastic limit, therefore, 
the operation gives a valuable quality to the steel. 

Effect of Manipulations Common to Constructive Processes; 
Also Punched, Drilled and Reamed Holes. 

The shop treatment of steel must in some respects be 
peculiar to that metal and different from that which 
characterizes the manufacture of wrought-iron bridge mem- 
bers. While the processes of punching and shearing may 
not be vSpecially injurious to comparatively thin plates and 
shapes of low steel and of the lower carbon grades of 
mild steel (perhaps up to a limit of 65,000 pounds per square 
inch) they are sufficiently injurious to heavier sections and 
to the higher grades of steel to necessitate the avoidance 
of their effects. If punches and dies are kept in good sharp 
condition, as they should be, the prejudicial effects are 



340 TENSION. [Ch. VII. 

lessened. The effect of a punch, however, under the best 
conditions of operation is not to make a smooth-sheared 
surface, but one of somewhat ragged or serrated character 
in which incipient cracks are started and which may be 
continued indefinitely into the interior of the metal unless 
some curative procedure is employed. 

It has been found by actual test that the region affected 
by the punch or by the jaw of the shear extends but a 
short distance from, the cutting-edge of the tool. Within 
that region, however, the metal is much hardened and the 
loss of ductility and elevation of elastic limit is due to that 
hardening. The decreased ultimate resistance is probably 
due to the violent disturbance of the molecules and the 
resulting minute fissures in the metal within the same 
region. In riveted work, the prejudicial effect is therefore 
removed by reaming the punched hole to a diameter about 
I inch larger than made by the punch. This removes a 
thin ring of injured metal about -^ of an inch thick, and 
it is found sufficient for the purpose. 

In large and heavy work it has come to be the practice 
by the best shops to make drilled holes in. which cases no 
question of the injury of metal can arise. The use of the 
drill leaves a sharp edge at each surface of the plate which 
tends to produce a shearing effect upon the corresponding 
rivet sections. Some specifications require this to be over- 
come by a quick application of a proper tool to remove the 
sharp edge. 

The general effects of the cutting edge of the shear 
are precisely the same as those of the punch, as the opera- 
tion in each case is a shear. Hence, if sheared edges 
are planed off to a depth of one-sixteenth to one-eighth 
of an inch, the injured metal will be entirely removed. 
The hardening effects of both shearing and punching 
may also be removed by the process of annealing, although 



Art. 58.] STEEL. ■ 341 

less effectually than by reaming and planing. As naturally 
would be inferred by experience in punching, higher steel 
and thicker plates are more injuriously affected by shear- 
ing than low steels and thinner plates. 

In consequence of the irregular edge of a large sheared 
plate, bridge specifications frequently require that at 
least one-quarter of an inch of metal shall be removed 
from the edge of such plates by planing. 

Steel seems to be very sensitive to the effects of hammer- 
ing or working at what is termed a " blue heat." Con- 
sequently it is necessary to heat the rivet to such a tem- 
perature as will enable the operation of heading to be 
completed before the rivet cools to the blue stage. A 
bright red or yellow heat is requisite for good work, and 
the rivet should be held under a pressure of fifty or sixty 
tons per square inch of the shaft section until the metal 
has timxC to flow throughout the rivet length and thus 
completely fill the hole, otherwise the upsetting will be 
complete at and in the vicinity of the rivet -heads only. 
An additional advantage in holding the rivet under the 
greatest pressure of the riveter for a short time is the 
fact that the rivet becomes cool enough to prevent the 
separation of the plates. 

The forging of steel requires unusual skill and ex- 
perience. When a piece has been heated to a proper 
temperature it should be kept under work tmtil it has 
fallen in temperature to a proper point to secure all 
the advantages of working, but of course not below 
red heat. The forging should be done with a hammer 
whose weight is suitably proportionate to the mass to be 
forged. If the hammer is too light, the result will be a 
surface effect only, with the interior but little changed. 
Pressure forging, with appropriate facihties for attaining great 
pressures, is probably capable of producing the best results. 



342 TENSION. [Ch. VII. 

The operation of annealing, particularly as applied 
to full-size bars, is one of great importance in the manu- 
facture of structural steelwork. The metal is heated as 
tmiformly as possible, so that tmdue stresses will not be 
developed, to a bright cheny-red, corresponding probably 
to about iioo or 1200 degrees Fahr., and then allowed 
to cool gradually. By this means any internal stresses 
that may have been produced by the process of forging, 
or any other shop manipulation, are eliminated. The 
metal is sufficient^ softened at the highest temperature 
to allow the molecules to adjust themselves to a condition 
of essentially no stress, and if the cooling is gradual the 
internal stresses will not be re-developed . 

Change of Ultimate Resistance, Elastic Limit and Modulus 
of Elasticity by Rete sting. 

It has been observed from the earliest experiences in 
testing steel and wrought iron that if a piece of material 
be subjected to an intensity of tensile stress higher than the 
elastic limit, thus producing permanent stretch, the ultimate 
resistance will be materially increased, although the duc- 
tility is generally decreased. Sufficient investigation has 
not even yet been undertaken to gage the full significance 
of such phenomena, but enough has been done to show 
some important results. 

It is yet uncertain whether an indefinitely long rest may 
not diminish to some extent at least the enhanced ultimate 
resistance of a piece of metal stressed beyond the elastic 
limit. Professor Bauschinger made some investigations in 
this special field many years ago which indicate that the 
elastic limit is considerably decreased by immediate retest- 
ing, but that such a decrease does not take place if a 
period of at least twenty -four hours or possibly more elapses 
before retesting. Some tests indicate that the elastic limit 



]Art. 58. STEEL. 343 

may be much increased even by suitable periods of rest 
between applications of loading. 

The yield point appears to be raised materially by re- 
testing, and the same observation as already indicated is 
equally applicable to the ultimate resistance. 

Fracture of Steel. 

The character of steel fractures has been incidentally 
noticed, in some cases, in the different tables. 

If the steel is low, or if some of the higher grades are 
thoroughly annealed, the fracture is fine and silky, pro- 
vided the breakage is produced gradually. In other 
cases the fracture is partly granular and partly silky, or 
wholly granular. 

In any case a sudden breakage may produce a granular 
fracture. 

The Effects of Chemical Elements on the Physical Qualities 

of Steel. 

Anything more than a meagre statement of the influ- 
ence of the chemical composition of steel on its physical 
properties is obviously out of place here, but a knowledge, 
however slight it may be, of the influence of certain ele- 
ments on those properties is so essential to the engineer 
in his structural work that attention should at least be 
called to it. 

Although other elements exert highly important influ- 
ences upon the resisting qualities of steel, carbon is tin- 
doubtedly the most prominent hardener. The effect 
of a given percentage of carbon, at least within certain 
rather wide limits, is to give greater toughness and resist- 
ing qualities to steel with less concurrent brittleness than 
any other contained element. It is made, therefore, 
the basis of classification of structural steel, the low steels 
being low in carbon and the high steels high in carbon. 



344 TENSION. [Ch. VII. 

The metal manganese also gives to steel some advan- 
tageous qualities. At the present time it seldom enters 
steel to an amount less than .5 per cent., nor more than 
about I per cent. Its presence seems to confer the capacity 
of resisting the effects of high temperatures in shop pro- 
cesses. Metal low in phosphorus and sulphur appears 
to require less manganese than that which is higher in. 
those impurities. It has been found that the influence of 
manganese upon steel depends in a rather extraordinary 
manner upon its amount. If the content reaches 1.5 
or 2 per cent, steel becomes practically worthless on 
account of its brittleness, but when a content of 6 or 7 
per cent, of manganese is reached, the metal becomies 
extremely hard and acquires to a high degree the property 
of toughness by quenching in water without becoming 
much harder. 

When steel is alloyed with more than about 7 per 
cent, of manganese, manganese -steel is the product, which, 
in its natural state, may have an ultimate tensile resist- 
ance running from 74,000 to OA^er 116,000 pounds per 
square inch. When quenched in water the ultimate 
tensile resistance of the same mictal mxay run from about 
90,000 pounds per square inch up to nearly 137,000 pounds 
per square inch. Before quenching the final stretch 
ranged from i to 4 or 5 per cent., and after quenching 
from 4 to 44 per cent. The preceding figures belong to 
a range in manganese from about 7 per cent, to over 19 
per cent, concurrently with carbon from about .61 per 
cent, up to 1.83 per cent. This metal is an interesting 
alloy, but is never used in structural engineering work. 

Opinions vary much as to the influence of silicon 
on steel, but it seems now to be well established that 
that influence within the limits ordinarily found is of 
minor consequence, or at least not prejudicial to either 



Art. 58.] STEEL. 345 

resistance or ductility. In structural steel it usually 
ranges from less than .03 to .05 per cent., while in rail 
steel it may run as high as .3 per cent. In some excellent 
tool-steel it may run even from .2 to .75 per cent. 

Sulphur is an impurity carrying with it highly preju- 
dicial effects. It essentially injures metal for rolling, as 
it makes the steel liable to crack and tear at the usual 
temperatures found between the rolls. It also diminishes 
capacity to weld. Its effects may, to some extent, be 
overcome b}^ the presence of manganese and by proper 
care in heating. It is, however, highly prejudicial as 
an element and is usually kept below about .04 per cent. 

Of all the objectionable elements found in steel, phos- 
phorus has the position of prim.acy. Although it is a 
hardener which may increase the ultimate resistance 
to som.e extent, it produces brittleness and diminishes 
most materially the capacity to resist shock, and it is 
one of the chief purposes of the best methods of steel 
production to reduce phosphorus to the lowest practicable 
limit. Its effects are sometimes erratic, being occasionally 
found in excess in apparently good material. In structural 
steel it is seldom permitted to nm over .08 per cent., and 
in the basic processes of manufacture it frequently falls 
to .03 or .04 per cent. 

The presence of .1 to .25 per cent, copper appears to 
have no deleterious eft'ect upon steel and may even be 
beneficial. As high as i per cent, of copper has been 
foimd in steel without serious effects where sulphur was 
low. 

Aluminum steel is an alloy containing at times as 
high as 5 to 6 per cent, of aluminum. The effect of alumi- 
num on ultimate resistance does not seem to be prejudicial, 
nor, again, is it of any special advantage; nor does it 
act seriously upon the ductility until its amount approaches 



346 



TENSION. 



[Ch. VII. 



about 2 per cent, or more. On the whole it does not 
seem to be a valuable element for steel. 

There are other special alloys such as tungsten and 
chromium steel. They are used for the special pj'^rposes 
of tools on account of their hardness, which is so extreme 
that neither quenching nor tempering is required. They 
do not, however, enter into structural use. 

Art. 59. — Copper, Tin, Aluminum, and Zinc, and their Alloys — 
Alloys of Aluminum — Phosphor-bronze — Magnesium. 

Anything like a complete knowledge of the physical 
properties of the alloys of copper, tin, aluminum, zinc, etc., 
is still lacking, although many investigations have been 
made in the past by the late Prof. R. H. Thurston and 
others, while other investigations are still in progress. The 
character of many of these alloys changes so radically for 
different proportions of the constituent elements and under 
different conditions of heat and other treatment that the 
results of tests are as varied as the relative amounts of 
the constituents and the physical conditions which attend 
the tests. Some of the results which follow belong to the 
earlier work of Prof. Thurston, but as they exhibit the same 
physical qualities as the corresponding alloys now used and 
as the later investigations do not cover the same field, they 
possess real value and are retained. 

Table I gives the tensile coefficients of elasticity {E) 
of copper and the alloys indicated as determined by Prof. 
Thurston. 

Table I. 



Metal. 



Authority. 



Remarks. 



Gun-bronze. . 

Alloy 

Alloy 

Tobin's alloy. 
Copper 



Thurston i 11,468,000 Copper, 0.90; tin, o.io (nearly). 

" 13,514,000 Copper, 0.80; zinc, 0.20. 

" I 14,286,000 Copper, 0.625; zinc, 0.375. 

" I 4,545,000 Composition, below table. 

" ■ I 9,091,000 Cast metal 



Art. 59. 



COPPER, TIN, ALUMINUM, ZINC, ETC. 



347 



Tobin's alloy is a composition of copper, tin, and 
zinc, in the proportions (very nearly) of 58.2, 2.3, and 
39.5, respectively. The value of E for this metal, and 
those for the two preceding and one following it, are 
calculated for small stresses and strains given by Prof. 
Thurston in the " Trans. Am. Soc. Civ. Engrs.," for Sept., 
1881. 

There will also be found in Tables VIII, IX, X and 
XI coefficients of elasticity for alumiinum-zinc, aluminum 
magnesium, and other alloys, and for magnesium, alumi- 
num, and zinc. 

Table II. 

CAST TIN. 



p- 


E. 


; p- 


E. 


1,950 
2,360 
2,580 


3,147,000 
472,000 
172,000 


3,200 
4,000 

Broke at 


96,400 
41,540 
4,200 lbs. 



Table III. 
CAST COPPER. 



p- 


E. 


p. 


E. 


800 

2,000 
4,000 

8,000 


10,000,000 
9,091,000 
9,091,000 

14,815,000 


12,000 
13,600 
16,000 
22,000 


18,750,000 

8,193,000 

2,235,000 

137,000 



Broke at 29,200 lbs. 

The values of E (stress over strain) for different inten- 
sities of stress (pounds per square inch) for cast tin, cast 
copper, and Tobin's alloy, are given in Tables II, III, 
and IV. 



348 



TENSION. 



[Ch. VIT. 



" /?'* is the intensity of stress in pounds per square inch, 
at which the ratio E exists. 

Each of these metals is seen to give a very irregular 
elastic behavior. 

Tables TI, III, and IV are computed from data given 
by Prof. Thurston in the United States Report (page 
425) and " Trans. Am. Soc. Civ. Engrs.," already cited. 

Table IV. 

TOBIN'S ALLOY. 



p- 


E. 


P- 


E. 


2,000 


4,545,000 


18,000 


5,455,000 




4,000 


4,545,000 


24,000 


5,941,000 




6,000 


4,088,000 


30,000 


6,250,000 




8,000 


4,938,000 


40,000 


6,390,000 




10,000 


5,263,000 


50,000 


4,744,000 




14,000 


5,110,000 


60,000 


3,436,000 





Broke at 67,600 lbs. 



Ultimate Resistance and Elastic Limit. 

Table V is abstracted from the results of the experi- 
ments of Prof. Thurston as given in the " Report of the 
U. S. Board Appointed to Test Iron, Steel, and other 
^letals," and *' Trans. Am. Soc. of Civ. Engrs.," Sept., 
1 88 1. The composition of the various alloys was as 
given in the table, which also contains results for pure 
copper, tin, and zinc. All the specimens were of cast 
metal. 

The mechanical properties of the copper-tin-zinc alloys 
have been very thoroughly investigated by Prof. Thurston 
("Trans. Am. Soc. of Civ. Engrs.," Jan. and Sept., 1881). 
As results of his work he has found that the ultimate 



Art. 59. 



COPPER, TIN, ALUMINUM, ZINC, ETC, 
Table V. 



349 



Percentage of 


Pounds Stress per 
Square Inch at 


Per Cent., Fina^ 


Cupper. 


Tin. 


Zinc. 


Elastic 
Limit. 


Ultimate 
Resistance. 


Stretch. 


Contrac- 
tion. 


100 
100 
100 

90 

80 

70 

62 

52 

39 

29 

21 

10 

00 

00 

00 

Gun 
90 
80 
62.5 

58.2 
100 
90.56 
81.91 
71 . 20 
60.94 

58.49 
49.66 
41-30 
32.94 
20.81 
10.30 
0.0 
70.0 
57-50 
45-0 
66 2=; 


00 

00 

00 

10 

20 

30 

38 

48 

61 

71 

79 

90 
100 
Queensl'd 
100 
Banca 
100 
Bronze 

10 

00 

00 

2-3 

0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 

8. 75 
21.25 

23-75 
23-75 
2.30 
50.00 
10.00 
20.00 


00 

00 

00 ' 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 
20 

37-5 
39-5 
0.0 
9.42 
17.99 
28.54 
38.65 
41 . 10 

50.14 

58.12 
66.23 
77-63 
88.88 
100.00 
20.25 
21.25 

31.25 
10.00 
39-48 
40.00 
30.00 
15.00 


11,620 
11,000 
14,400 
15,740 


19,872 
12,760 
. 27,800 
26,860 
32,980 

5,585 
688 

2,555 
2,820 
1,648 
4,337 
6,450 
3,500 

2,760 

3,500 

31,000 
33,140 
48,760 
67,600 
29,200 


0.05 

0.005 

0.065 

0.037 
0.004 


10. 

8.0 

150 

13-5 
00.0 
00.0 
00.0 
00.0 
00.0 
00.0 
00.0 
15-0 
75-0 


5,585 
. 688 

2,555 
2,820 








0.07 
0.36 


3,500 
1,670 


2,000 
10,000 , 


0.36 

4-6 
32.4 
31.0 

4-0 

7-5 


47 
86.0 








29 -5 

8.0 

16.0 






10,000 

9,000 
16,470 
27,240 
16,890 

3,727 

1,774 

9,000 
14,450 

4,050 
18,000 (?) 

1,300 

2,196 

3.294 
30,000 (?) 

5,000 (?) 
21,780 (?) 


32,670 

30,510 

41,065 

50,450 

30,990 

3,727 

1,774 

9,000 

14,450 

5,400 

31,600 

1,300 

2,196 

3,294 
66,500 

9,300 
21,780 

3.765 


31-4 
29.2 
20.7 
10. 1 
50 


43 
38.0 
28.0 
17.0 
11-5 






0. 16 

039 
0.69 

0.36 


0.0 
0.0 
0.0 
0.0 






58.22 
10.00 
60.00 
65.00 


3-13 
0.7 

0.15 


7.0 
0.0 
0.0 









3 so 



TENSION. 

Table V. — Continued. 



[Ch. VII. 



Percentage of 


Pounds Stress per 
Square Inch at 


Per Cent., Final 


Copper. 


Tin. 


Zinc. 


Elastic 
Limit. 


Ultimate 
Resistance. 


Stretch. 


Contrac- 
tion. 


70.00 
75.00 
80.00 
55 00 
60.00 
72.50 
77-50 
85.00 


10.00 
5-00 

10.00 
0.50 
2.50 
7.50 

12.5 

12.5 


20.00 
20.00 
10.00 
44 50 
37-50 

2.00 
10.00 

2.50 


24,ooo(?) 

I2,000(?) 

I2,000(?) 

22,000 

22,000 

1 1 ,000 

20,000 

I2,000(?) 


33,140 
34.960 
32,830 
68,900 
57,400 
32,700 
36,000 
34,500 


0.31 
3-2 

1.6 
9-4 
4.9 
3-7 
0.7 
1.3 


5-4 
4.0 

25.0 
6.6 

II. 
0.0 
30 



The. values of the elastic limit in the lower part of the table were not at 
all well defined. 

tensile resistance, in pounds per square inch, of " ordinary 

bronze, composed of copper and tin, as cast in the 

ordinary course of a brassfounder's business," may be 
well represented by 

T^ = 30,000 + I ,oooi( ; 

** where t is the percentage of tin and not above 15 per 
cent." 

" For brass (copper and zinc) the tenacity may be 
taken as: 

7^ = 30,000 + 500.^, 

where z is the percentage of zinc and not above 50 per 
cent." 

He found that a large portion of the copper-tin -zinc 
alloys is worthless to the engineer, while the other, or 
valuable portion, may be considered to possess a tenacity, 
in pounds per square inch, well represented by combining 
the above formulae as follows: 

Tzf = 30,000 + 1 ,000/ -f 5000. 

These formulae are not intended to be exact, but to 



Art. 59.] COPPER, TIN, ALUMINUM, ZINC, ETC 351 

give safe results for ordinary use within the limits of the 
circumstances on which they are based. 

Prof. Thurston found the " strongest of the bronzes" 
to be composed of: 

Copper o 55 .0 

Tin 0.5 

Zinc 44. 5 

100.00 

This alloy possessed an ultimate tensile resistance of 
68,900 pounds per square inch of original section, an 
elongation of 47 to 51 per cent, and a final contraction 
of fractured section of 47 to 52 per cent. 

The first and sixth alloys of copper, tin, and zinc, in 
Table V, are called by Professor Thurston ''Tobin's alloy." 
" This alloy, like the maximum metal, was capable of 
being forged or rolled at a low red heat or worked cold. 
Rolled hot, its tenacity rose to 79,000 poimds, and when 
moderately and carefully rolled, to 104,000 potmds. It 
could be bent double either hot or cold, and was found 
to make excellent bolts and nuts." 

As just indicated for the particular case of the Tobin 
alloy, the mianner of treating and working these alloys 
exerts great influence on the tenacity and ductility. 

Professor Thurston states: "brass, containing copper 
62 to 70, zinc 38 to 30, attains a strength in the wire mill 
of 90,000 pounds per square inch, and somj.etimes of 100,000 
poimds." 

All of Professor Thurston's specimens were what may 
be called "long" ones, i.e., they were turned down to 
a diameter of 0.798 inch for a length of five inches, giving 
an area of cross-section of 0.5 square inch. 



35^ 



TENSION. 



[Chrvil. 



Alloys of Aluminum. 

Prof. R. C. Carpenter, of Cornell University, in the 
transactions of the Am. Soc. Mech. Engrs., vol. xix, has 
reported a ntimber of interesting and valuable tests of 
alloys of aluminum, as well as tests of pure magnesium. 

Table VI . 
ALLOYS OF GREATEST RESISTANCE. 



Percentage of 


Ultimate 
Resistance, 

Lbs. per 
Square Inch. 


Specific 
Gravity. 


Per Cent, of 


Aluminum. 


Copper. 


Tin. 


Final Stretch. 


85. 
6.25 
5. 


7-5 

87-5 

5- 


7-5 

6.25 
90. 


30,000 
63,000 
11,000 


3.02 

7-35 
6.82 


4- 

3.8 

10. 1 



The greater part of the results for t?ie aluminum-tin- 
copper alloys are given in Table VII, but the composition 
of those giving the greatest ultimate resistances are ex- 
hibited in Table VI. It will be observed that the highest 
ultimate resistance belongs to the alloy of greatest density 
but the alloy of least resistance has nearly as great density. 
The ductility of none of these alloys of greatest ultimate 
resistance is specially marked; indeed, the ductility is 
very low except in the case of the least ultimate resistance. 

The composition and corresponding elastic limits and 
ultimate resistances of aluminum-tin-copper alloys will 
be found in Table VII. Like all the aluminum alloys 
the specific gravity varies between wide limits, being 
low where there is much aluminum and high where there 
is little. The ductility is low in all cases except in that 
of pure tin or the alloy in which it appears to the extent 
of 90 per cent. There is in this table the usual wide range 
of physical qualities belonging to such a series of mixtures. 



Art. 59.] 



COPPER, TIN, ALUMINUM, ZINC, ETC. 



353 



Table VII. 
ALUMINUM ALLOYS. 



Composition. Per Cent. 


Ultimate Resistance, 










by Weight. 


Lbs. per Square Inch. 


Elastic Limit, 

Lbs. per 
Square Inch. 


Specific 


Final 

Stretch 

Per Cen 

in 6 












Gravity. 


Al. 


Tn. 


Cu. 


A, 


B. 






Inches. 






100 
90 


27,000 
40,815 


28,330 
42,038 


12,000 

13,832 


6 s 


6.5 
4.0 


5 


5 


7 


6 


10 


10 


80 


32,209 


34,200 


24,829 


6 


5 


0.8 


20 


20 


60 


1,966 


2,225 


* 


5 


7 




30 


30 


40 


849 


1,077 


* 


5 


05 




40 


40 


20 


4,800 


5,672 


* 


4 


91 




100 






15,000 


14,316 


6,432 


2 


67 

82 


5.6 

3- 


90 


5 


5 


15,476 


17,070 


8,227 


2 


80 


10 


10 


18,580 


2 1 , 1 40 


13.329 


3 


09 


1.2 


60 


20 


20 


4,416 


5,950 


* 


3 


53 


• 3 


40 


30 


30 


915 


1,123 


* 


4 


4 




20 


40 


40 


2,221 


2,622 


* 


5 


21 






100 




3,505 
11,582 


3,933 
10,418 




7 
6 


" 


35.51 
10. 15 


5 


90 


5 


4,823 




77 


10 


80 


10 


5,999 


5,922 


2,988 


6 


24 


I . I 


20 


60 


20 


1,198 


1,200 


* 


5 


55 




30 


40 


30 


993 


, 961 


* 


4 


96 




40 


20 


40 


3,798 


3,997 


* 


4 


48 





A. Results of first melting, B. Results of second melting. 
Test pieces 6 in. between shoulders, diam. J inch. 
* Could not be turned in the lathe. 

The results in this table were obtained by Messrs. Geb- 
hardt and Ward, at the testing laboratory of Sibley Col- 
lege of Mechanical Engineering, Cornell University, in 
1896. 

The physical properties of alurninnm-zinc alloys, in- 
cluding those metals unalloyed, are equally fully given 
in Table VIII, as well as the values of the coefficients of 
elasticity. There is not as wide variation of results in this 
table as in Table VII, although there is a considerable 
range of ultimate resistance, especially if the results for 
unalloyed zinc be included. It will be observed that 
this table also includes the intensity of stress found in 



354 



TENSION. 



[Ch. VII. 



Table VIII. 
ALUMINUM-ZINC ALLOYS. 



Percentage. 



Alumi- 
num. 



90 
90 

85 
85 
80 
80 
75 
75 
70 



65 
60 
60 
50 
50 
25 
25 



Zinc. 



O 

o 
10 
10 

15 
15 

20 
20 

25 
25 
30 
30 

33 
35 
40 
40 
50 
50 
75 
75 



Specific 
Gravity. 



2 


67 


2 


67 


2 


77 


2 


74 


2 


918 


2 


918 


2 


998 


2 


975 


3 


15 


3 


14 


3 


191 


3 


24 


3 


326 


3 


471 


3 


57 











7.19 



Ultimate 
Resist- 
ance, 

Lbs. per 
Sq. In. 



14,460 
16,750 
17,940 



18,100 
21,850 



22,940 



24,400 
23,950 



19,770 
19,300 
19,060 

13,175 
14*150 

2,522 



Transverse 

Tests. 
Maximum 
Fibre 
Stress 
Lbs. per 
Sq. In. 



14,500 
14,150 
18,950 

28,091 



34,600 



45,080 
43,200 
41,200 



40,350 
38,100 
39,850 
25,500 



7,556 



Coeflficient 

of 
Elasticity. 



6,535,000 



7,710,000 
9,260,000 



9,110,000 
8,210,000 
8,178,000 



8,540,000 
8,500,000 
8,670,000 
6,680.000 



Remarks. 



Shrinkage uneven. 
(I << 

Shrinkage uneven 

It (t 

Shrinkage even. 

Poor specimen. 



i Elongation of all the 
specimens less than 
I per cent. 



Note. — The experimental results given in Table IX are those of Messrs. 
Hunt and Andrews, obtained at Sibley College of Mechanical Engineering, 
Cornell University, in 1894. 

Table IX. 
TENSILE TESTS OF MAGNESIUM— CAST METAL. 



Number of 
Test Piece. 


Diameter. 


Ultimate 
Resistance, 

Lbs. per 
Square Inch. 


Elastic Limit, 

Lbs. per 
Square Inch. 


Final 
Extension, 
Per Cent. 


Coefficient of 
Elasticity. 


I 


•433 
.433 
.442 
■ 435 
.424 

•432 


23,800 
22,050 
20,900 
19,500 
24,800 
■22,500 


8,800 


4-2 


2,040,000 


2 


T, 860,000 


3 

4 

5 

6 


10,780 
8,400 
7,090 


1.8 
2.5 
31 
2.3 


2,060,000 
1,830,000 
1,930,000 









Art. 59.1 



COPPER, TIN, ALUMINUM, ZINC, ETC. 



355 



Table X. 
ALLOYS OF ALUMINUM AND MAGNESIUM 



Number of 
Test Piece. 


Percentage of 
Magnesium. 


Specific 
Gravity. 


Ultimate 
Resistance, 

Lbs. per 
Square Inch. 


Elastic Limit, 

Lbs. per 
Square Inch. 


Coefficient of 
Elasticity. 


I 



2 

5 
10 

30 


2.67 
2.62 
2.59 
2.55 
2. 29 


13,685 
15,440 
17,850 
19,680 
5,000 


4,900 

8,700 

13,090 

14,600 


1,690,000 
2,650,000 
2,917,000 
2,650,000 


2 

'3. 


4 


c , . 







the extreme fibres of beams subjected to transverse load- 
ing. Although these values are not required at this 
point, it is more convenient to insert them here and refer 
to them in the article devoted to the flexure of such beams. 
The sizes of tlie specimens subjected to transverse load- 
ing are not given, but they Vv^ere small. 

Table XL 



Character of Alloys. 


Resistance, Pounds per Square Inch. 




6 


ComposiLlcn and Remarks. 


Tension. 


Transverse. 




Al. 


Cu. 


Tin. 




Elastic. 


Ulti- 
mate. 


Elastic. 


Ulti- 
mate. 




I 
2 

3 


/o 
100 

93 

75.7 


07 
/o 


% 




4,000 
6,250 


12,055 
18,555 
35,075 


2,.345 
9,000 




t^ 


7 
3 






25,250 
23,420 


6 




20% zinc, 1.3 man. 








4 
5 
6 
7 
8 

9 
10 


100 
100 
98 
98 
96 
96 
96.5 






I inch bars. ...... 

f " " 


12,500 


17,185 


17,154 
18,870 
13,720 
18,870 
22,300 
30,880 
26,313 












i 


2 
2 
4 
4 
2 




I " " 

f " " 


9,000 


18,647 




S 


I " " 

3 «< i< 


16,000 


23,045 






i|% chromium. 


19,000 


26,310 






II 
12 
13 
14 
15 


98 
96 

94 
92 
90 




2 

4 

6 

8 

10 




2,150 
2,400 
2,250 
2,000 
1,750 


8,622 
9,565 
9,315 
7,270 
7,352 












. 


H >, 








S3 








0^ 

















356 



TENSION. 



[Ch. Vli. 



The experimental results given in Tables IX and X 
were also established at the testing laboratory of Sibley 
College of Mechanical Engineering of Cornell University. 
The tests were made by Messrs. Marks and Barraclough, 
graduate students in 1893. Table IX gives results for 
pure magnesium, including the coefficients of elasticity 
and the final stretch, while Table X exhibits the results 
for alloys of aluminum and magnesium, the per cent, 
of magnesium being shov/n in one of the columns, the 
remaining per cent, being aluminum. The ultimate resist- 
ances given in Table IX show that magnesium is a metal 
of considerable tensile resistance, especially in comparison 
with its density, its specific gravity being but 1.74, that 
of aluminum being 2.67. 

Table XI exhibits the elastic limits and ultimate 

Table XL — Continued. 



Final, stretch 
Per Cent. 
(Tension 
Pieces). 


Final 

Contraction, 

Per Cent. 

(Tension 

Pieces). 


Hardness 
(Relative). 


specific Gravity 
of specimen. 


Coefficient of Elasticity. 
Lbs. per Square Inch. 


Ten- 
sion. 


Trans- 
verse. 


Tension. 


Transverse. 


5.62 

I. GO 

.15 


iO.93 
3.08 
1.77 


3.61 

12.87 
35.56 


2.670 
2 . 830 
3II7 


2.654 
2.810 

3.055 


8,385.000 

11,115,000 

9,685,000 


8,440,000 
8,065,000 
8,060,000 


8.49 


38.30 


7.12 

6.94 

6.79 

12.30 

12 .42 

13.35 
14.09 


2.710 

2.715 
2.725 
2 .756 
2.774 
2.773 
2.759 


9,780,000 


10,110,000 

10,000,000 

10,330,000 

Q 600 000 


19.49 


39.02 


9,505,000 


3.62 


10.10 


10,440,000 


10,595,000 

10,070,000 

9,813,000 


I-3I 


9.78 


9,850,000 


4.00 
5.38 
5.19 
3.06 

3.87 


8.64 
6.86 
7-97 
5.41 
8.89 


3-71 
3.74 
3-49 
3-33 
3.09 


2 . 689 

2.739 
2.771 
2.804 
2.856 


5,435,000 
6,210,000 
5,035,000 
5,175,000 
6,675,000 





Art. 59.] 



COPPER TIN, ALUMINUM, ZINC, ETC. 



357 



resistances of all the different alloys shown in the table, and 
in the conditions also exhibited by the table, i.e., whether 
cast or rolled. There are also given coefficients of elasticity 
for both tension and transverse tests, as well as elastic 
limits' and ultimate stresses (intensities) in the extreme 
. fibres of small beams, to which reference will be made 
in the article devoted to transverse resistance. 

It will be observed that both the elastic hmits and 
the ultimate resistances of Table XI are found within 
the range exhibited by the results already shown in the 
preceding tables. 

If desired, diagrams can readily be constructed from 
the results of each table which will show the variations 
of physical quantities corresponding to the variations of 
composition of the alloys. 

In 1895 the Fairbanks Company tested at their New 
York office four specimens of Tobin bronze manufactured 
by the Ansonia Brass and Copper Co., with the following 
results. 

ROLLED TOBIN BRONZE PLATES— SPECIMENS 8 INCHES LONG. 



Specimen, 
Inches. 


Resistance in Pounds per Sq. Inch. 


Per Cent., Final 


Elastic. 


Ultimate. 


Stretch. 


Contraction. 


1X.185 
1X.185 
1X.25 
1X.25 


51,350 
51,350 
56,000 
56.450 


78,920 
78,810 
79,200 
79,640 


20.5 
17-5 
17.5 
16.25 


45-4 
44.33 
43.2 
40.72 



Alloys of Aluminum and Copper. 

In 1907, Prof. H. C. H. Carpenter, M.A., Ph.D., and 
Mr. C. A. Edwards, made their Eighth Report on alloys 
of aluminum and copper to the Alloys Research Committee 
of the Institution of Mechanical Engineers of Great Britain. 



358 



TENSION. 



[Ch. VII. 



This alloy is known as ** aluminum bronze " or " gold." 
These investigators made over a thousand tests in tension 
and torsion and in other ways, including heat treatment for 
both cast and rolled material, The investigation is one of the 
most important ever made with this class of alloys. Out 
of the great number of tests contained in the report, Table 
XII has been selected as sufficiently typical for the purpose 
of conveying a correct impression of the character of the 
work done. 



Table XII. 

The percentage of aluminum only is given in the Table, as the alloy is of 
aluminum and copper, the remaining percentage being copper. 





Al. 


Yield Point 


Ult. Resist. -p 




Elongation 


No. 


per cent. 


lbs. per sq. 
in. 


lbs. per sq. ^^ 
in. 


atio. 


in 2 inches 
per cent. 


I 


O.I 


8,512 


25,760 


33 


46 


2 


I 


o6 


6,720 


30,020 


22 


52 


3 


2 


I 


7,616 


30,240 


25 


53-5 


4 


2 


99 


8,512 


32,480 


26 


60 


5 


4 


05 


7,840 


37,410 


21 


83 


6 


5 


07 


9.632 


40,540 


24 


75 


7 


5 


76 


10,752 


39,870 


27 


67 


8 


6 


73 


10,752 


41,780 


26 




9 


7 


35 


14,784 


47,710 


31 


71 


10 


8 


12 


17,248 


55,800 


31 


58 


II 


8 


67 


21,952 


62,944 


35 


48 


12 


9 


38 


21,728 


68,050 


32 


36.2 


13 


9 


9 


25,312 


71,010 


36 


21.7 


14 


10 


78 


31,584 


59,750 


48 


9.0 


15 


II 


73 


31,360 


56,960 


55 


5 


i6 


13 


02 


44,240 


44,240 . . . 




I 






t^'t, t" 







It will be observed that the specimens were of cast, 
metal. While the rolled specimens give somewhat higher 
ductility, in the main there is much less difference than 
would probably have been anticipated. Although the 
elastic ratio, i.e., the ratio of the elastic limit over the 
ultimate, is somewhat higher for the rolled specimens, the 
difference on the whole is not great, except in a compara- 



Art. 59.1 COPPER, TIN, ALUMINUM, ZINC, ETC. 359 

tively few instances. In fact, the differences in results 
found by the investigators between the cast and rolled 
metal are much smaller than might have been expected. 

The authors of the report state, among other obser- 
vations : 

" (a) The limit of industrially serviceable alloys must 
be placed at 11 per cent, of aluminum. For most purposes 
the limit might be put at 10 per cent., beyond which there 
is a rapid fall of ductility with no rise of ultimate resist- 
ance. . . . 

" (b) Between these limits the alloys fall into two 
classes: i. those containing from o to 7.35 per cent, of 
aluminum: 2. Those containing from 8 to 11 per cent, of 
aluminum. Class i represents material of apparently low 
yield point and moderate ultimate stress, but of very good 
ductility. The introduction and further addition of alumi- 
num causes a gradual increase of strength but hardly affects 
the ductility. It is true that as regards the steadiness of 
the ductility this has only been establivshed for the rolled 
bars. But the sand and chill castings have shown the same 
kind of variations as the rolled bars in all the properties 
examined. . . . 

" Into Class 2 come alloys of relatively low yield point 
but good ultimate stress. From 8 to 10 per cent, of alumi- 
num the ductility is also good. ..." 

To gain an adequate idea of the physical properties of 
the various grades of this alloy of aluminum and copper 
requires a full scrutiny of the entire report. 

Bronzes and Brass Used by the Board of Water Supply of 
New York City. 

In the construction of the Additional Catskill Water 
Supply for the city of New York by the Board of Water Sup- 
ply a large amount of bronze castings and rolled bronze, as 



-,6o TENSION. [Ch. VII. 

well as brass, was used for a great variety of large and 
small articles varying from a number of tons in weight each 
to a few pounds, such as small bolts. The specifications 
prescribed that " Whenever the term. ' bronze ' is used in 
these Specifications in a general way or on the drawings, 
without qualification, it shall mean manganese or vanadium 
bronze or monel metal. . . . 

" The minimum physical properties of bronze shall, 
except as otherwise specified, be as follows: 
Castings: 

Ultimate tensile strength 65,000 lbs. per sq.in. 

Yield point 32,000 lbs. per sq.in. 

Elongation 25 per cent. 

Rolled Material: 

Ultimate strength 72,000 lbs. per sq.in. 

Yield point 36,000 lbs. per sq.in. 

Elongation 28 per cent. 

Rolled material, thickness above one inch: 

Ultimate strength 70,000 lbs. per sq.in. 

Yield point 35,000 lbs. per sq.in. 

Elongation 28 per cent." 

The modulus of elasticity E for tension and compression 
was about 14,000,000. 

The requirements of these specifications were even 
exceeded both in resistances and in ductility. Much trouble, 
how^ever, was experienced by the rolled metal exhibiting 
cracks and failures in articles large and small, in many cases 
even before put in place in the w^ork and subjected to duty. 
vSuch difficulties, however, were not experienced in castings. 
Investigations intended to discover the origin of these 
difficulties have not yet been completed, but they are prob- 
ably due to some feature of manipulation of material during 



Art. 59.1 COPPER, TIN, ALUMINUM, ZINC, ETC. 361 

processes of manufacture, including the treatment of the 
molten metal. 

Phosphor-Bronze . 

Phosphor-bronze possesses merit not only as a structural 
material on account of its high elastic limit and ultimate 
resistance, but also because it is a good anti-friction metal. 
Its elastic limit may be taken from 45,000 to 55,000 pounds 
per square inch and its iiltimate resistance from 50,000 to 
75,000 pounds per square inch, both values being given for' 
unannealed material. The same material as unc^nnealed 
wire with a diameter of one-tenth to one-sixteenth of an 
inch may give ultimate resistances var^dng from 100,000 
to 150,000 pounds per square inch, or if annealed not more 
perhaps than 50,000 to 60,000 per square inch. In the 
latter case, however, the final stretch may run from 3 c 
to 40 per cent. 

Bauschinger s Tests of Copper and Brass as to Effects of 

Repeated Application of Stress. 

The late Professor Bauschinger made some investiga- 
tions regarding the effect on elastic limit and yield point 
of repeated application of loading similar to those made 
on steel and wrought iron. The grade of brass used in his 
tests was called " red brass." 

With the exception of one case of brass the elastic limit 
and the yield point were both materially elevated by 
repeated application of loading, whether the repetition was 
made without a period of rest between two consecutive 
applications or not. Some repetitions were made immedi- 
ately and some after periods of 17I to 53 hours of rest. 

The effect on the modulus of elasticity was small and 
irregular, i.e., in some cases there was a small increase and 



362 TENSION. [Ch. VII. 

in others a small decrease and in some cases no material 
change. 

Art. 60. — Cement, Cement Mortars, etc. — Brick. 

The ultimate tensile resistance of cements and cement 
mortars depends upon many conditions. The two great 
divisions of cements, i.e., natural and Portland, possess very 
different ultimate resistances whether neat or mixed with 
sand, the latter being much the stronger. With given 
proportions of sand or neat, the ultimate resistances of 
cement mortar or cement will vary with the amount of water, 
used in tempering and with the pressure under which the 
moulds are filled. Again, the character of the sand used 
will obviously influence largely the tensile resistance of the 
mortar produced, and not only the degree of cleanliness, 
but the size of grain and the variety of sizes are elements 
which must be considered. It has also been maintained by 
some that silica-sand will give better results, other things 
being equal, than other sand. Finally, the shape of 
briquette used will affect the results to some extent. Fig. i, 
on page 370, shows the form of briquette recommended by the 
Committee of the American Society of Civil Engineers, and 
it is the form generally used in American practice. It is 
foreign to the purpose of this work to enter into the consider- 
ation of all these influences; they are ^ only mentioned to 
enable the few typical experimental results which follow 
to be interpreted properly. 

• As the fineness of grinding is an important quality of a 
cement, it is usually noted by stating the percentage of 
weight of tlie cement which either passes through or is 
retained ux.>on a sieve having a stated number of meshes 
per linear inch, which number squared gives the number 
of meslies per square inch. The sizes of the grains of sand 



Art. 60.] CEMENT, CEMENT MORTARS, ETC.— BRICK. 363 

used are graded in the same way. The " No." of a sieve 
to which reference may be made in what follows indicates, 
therefore, the number of meshes per linear inch. 

Modulus of Elasticity. 

In consequence of the fact that cement, mortars, and 
concrete begin to exhibit permanent stretch at compara- 
tively low tensile stresses there is a little uncertainty as to 
the value of the modulus of elasticity unless distinct state- 
ment is made of the intensities of stress at which those 
values are obtained, and whether the total stretch is 
used or that total less the permanent set. It is not possible 
to make such statement in connection with all the values 
which follow, except that thc}^ have been reached at low 
intensities of stress unless otherwise stated, and with 
elongations w^iich may be considered wholly elastic. Al- 
though cement mortars and concrete do not exhibit a per- 
fectly elastic behavior their stress-strain lines for intensities 
of stress even exceeding those used in practice are essentially 
straight and, on the whole, exhibit elastic properties at 
least equal to those of cast iron. 

Comparatively few tests have been made to determine 
either the tensile or compressive modulus of elasticity of 
cement, mortar and concrete, although that quantity is a 
most important element in the theory and design of much 
concrete work and reinforced concrete members. Mr. W. H. 
Henby of St. Louis, made a number of determinations of 
the tensile modulus of elasticity of Portland cement con- 
crete of 1-2-4, 1-2-5, 1-3-6, and 1-4-8 mixtures and gave 
the results in a paper read before the Engineers Club of 
St. Louis in 1900. He obtained values varying from less 
than 2,000,000 to 8,360,000. Other tests, however, indi- 
cate that values above perhaps 3,000,000 should not be 



364 TENSION. [Ch. VII. 

used. While higher values of the modulus of elasticity for 
rich mixtures of concrete may exist, the more important 
considerations of design usually bear upon work in which 
concrete must take serious loading when less than thirty 
days of age. 

For all these reasons it will seldom be advisable to take 
the modulus of elasticity of even as rich a mixture as i 
cement, 2 sand, and 4 broken stone higher than about 
2,500,000, and it will be seen later that in concrete steel 
w^ork w^here portions of a structure are liable to be loaded 
to a material extent within a comparatively short time 
after removal of the forms, it is the usual practice to consider 
the modulus as having a value of 2,000,000 only. These 
considerations are confirmed by the results of tests given 
below. 

Professor W. Kendrick Hatt, of Purdue University, 
in a paper read before the American Section of the Inter- 
national Association for Testing Materials, at its con- 
vention, 1902, gave the following values for the tensile 
coefficient of elasticity and ultimate tensile resistances of 
Portland cement concrete composed of i cement, 2 sand, 
and 4 broken stone at the ages of 25, 26, 28, and 33 days: 





Coefficient of Elasticity, 
Lbs. per Sq. in. 


Ultimate Tensile 

Resistance, 
Lbs. per Sq. In. 


jMaximuin 


2,700,000 
2,100,000 
1,400,000 


360 
280 


Average . ..... 


Minimum ... 







It will be found in discussing the compressive modulus 
of elasticity that both moduli probably acquire nearly 
their full value in about three months' time. It would 
appear that moduli do not increase in value with the lapse 
of time to the same extent as the ultimate resistance to 



Art. 60.] CEMENT, CEMENT MORTARS, ETC.— BRICK. 365 

compression, although conclusive data as to this point are 
not complete. 

Such tests as have been made show that the modulus 
of elasticity in tension or compression for cinder concrete 
should not be taken higher than about 1,250,000 for 1-2-5 
mixtures. Some tests show somewhat lower values and 
others values running over 2,000,000, but the latter results 
are too high for cinder concrete as ordinarily made and 
put in place. 

Ultimate Resistance. 

The ultimate resistances of neat Portland cement and 
mortar made with the same cement have been somewhat 
increased within the past half dozen years; but, upon the 
whole, those resistances as exhibited in the following 
tables are fairly representative of the best grades of cement 
used at the present time (191 5). The conditions of manu- 
facture are now so well controlled that a high 7 -day or 
28-day test cement may readily be produced; but that 
is not always desirable; the main purpose in masonry 
construction being rather the attainment of an ultimate 
resistance possibly less high under a short-time test but 
which continues to increase indefinitely. A cement show- 
ing a high ultim.ate resistance on a short-time test may not 
continue to increase its ultimate resistance satisfactorily, 
or that resistance may even recede for a time. 

The following tabular statement is of interest and value 
as indicating the character of the cement used in the con- 
struction of the first subway for the Rapid Transit Railroad 
in the City of New York. It will be observed that the 
ultimate resistances of both the neat cement and the mix- 
ture of I cement, 2 sand, are practically as found a dozen 
years later. The number of briquettes broken during the 
years igoo and 1901 was over 18,000. The average ulti- 



366 



TENSION. 



[Ch. VII. 



mate tensile resistances in pounds per square inch found 
by that series of tests of both Portland and natural cements, 
as given in the report of the Chief Engineer, are the 
following : 





Year. 


Neat Cement. 


Sand 2, Cement i* 




I Day. 


7 Days. 


28 Days. 


7 Days. 


28 Days. 


Portland: 

Average result 

Average result 

Spec requirements 


1900 
I901 


229 
300 


645 
400 

172 
215 

125 


714 
763 
500 

249 
322 
200 


276 
380 
200 

118 
218. 
100 


434 
525 
300 

215 
350 
150 


Natural: 

Average result 

Average result 

Spec reciuirements . 


1900 
190I 









* For natural cement a i cement i sand mortar was used. 



The results for the natural cement are of interest, as 
that material has at present (191 5) practically disappeared 
from use in consequence of the low prices for which Portland 
can be produced. 

Table I exhibits the results of tests of briquettes of 
different brands of domestic Portland cement as made in 
the testing laboratory of the Bureau of Surveys of the 
Department of Public Works of Philadelphia, Pa., for 
the year 191 2. This table gives the fineness of the cements 
in terms of the percentages by weight which were retained 
on sieves with 2500, 10,000 and 40,000 meshes per square 
inch; it also shows the amount of water used for the 
different mixtures, as well as the specific gravities of the, 
material. It will be observed that the briquettes were made 
of neat cement and of mortar with a mixture of i cement 
to 3 sand. The results, therefore, show the effect of the 
presence of sand on the ultimate tensile resistance of the 
matrix. The periods at which the briquettes were tested 
are the standard 24 hours, 7 days and 28 days. 



Art. 60.1 



CEMENT, CEMENT MORTARS, ETC.— BRICK. 



367 



Table I. 

Average Results of Tests of Portland Cement Made during 19 12 — Phila., Pa. 





'0 :;:^ 


Fineness in per 
cent. 


>> 




Tensile strength in pounds 
per square inch. 


Brand 


No. 
SO. 


No. 
100. 


No-. 
200. 


Neat 


1:3 




24 

hrs. 


7 

dys. 


28 

dys. 


7 

dys. 


28 

dys. 


Allentown. . . 
Alpha.. ..'... 

Atlas 

Bath 

Dexter 

Dragon ...... 

Edison 

Giant 

Lehigh 

Nazareth. . . . 
Northampton 

Paragon 

Penn Allen.. . 

Phoenix 

Saylor's 

Vulcanite. . . . 
Whitehall . . . 


816 

500 

168 

388 

582 

532 

630 

28 

2,026 

1.956 

28 

42 

514 

572 

1,984 

150 
1,162 


0.0 

0.0 
0.0 
0.0 
0.0 

O.I 
0.2 
0.0 
0.0 
0.0 
0.0 

0.3 
0.0 
0.0 

0.0 

0.0 

0.0 


3 
4 
4 
3 
2 

4 
2 

4 
3 
I 

3 

5 
4 
3 
3 
3 
3 


6 
8 

I 
4 
3 


7 

•3 

I 

9 

4 
3 
8 


I 
5 
5 


19.7 

23-5 
23.2 
20.5 
17.8 
20.9 
18.9 
22.8 

19.4 
16.8 
22. 1 
21.8 
23.2 
20.1 
21. 1 
21.7 
21.2 


3-174 
3. 161 

3-I5I 
3-130 
3.128 
3.106 

3-II4 
3.202 
3.172 
3-151 
3-138 
3.082 
3-156 
3.146 
3.127 
3-165 
3-155 


20.0 
19.9 
19.8 
20.3 

20.6 
21.6 

23.1 

20.0 

20.0 
20.6 
20.0 

23-3 
20.0 
20.7 
19.9 
20.0 
20.0 


377 
399 
480 

434 
434 
398 
261 
563 
363 
453 
497 
355 
446 
372 
277 
267 
429 


721 
701 
656 
710 

767 
704 

598 
672 

752 
776 

727 

644 

686 
670 
706 
717 
713 


797 
770 

741 
741 
820 
718 
670 

751 
812 
830 
812 
674 
735 
723 
801 
746 
759 


379 
367 
348 
384 
376 
370 
334 
366 
410 
403 
393 
328 

373 
364 
327 
376 
389 


499 
450 
430 
468 
450 
436 
412 
398 
498 
467 
445 
429 
475 
456 
436 
467 
477 



Table II shows the maximum, mean and minimum 
results of the tests of briquettes of various brands used by 
the Board of Water Supply of the City of New York during 
1 9 1 4 . During the past few years American Portland cement 
has been improved in uniformity of quality and fineness of 
grinding. These tests, therefore, show the latest results 
of the best practice in cement production and use. The 
tabulated values show the variations occurring in systematic 
testing of large quantities of cements at 7 -day and 28-day 
periods. The results are all in pounds per square inch and 
so arranged, as is evident, that in each vertical group of 
three in each column the highest value is the maximum and 
the lowest, the minimum, the mean occupying the middle 
position. 



368 



TENSION. 
Table II. 



[Ch. VII. 



Brand. 



Approx. 
Bbls. 



Alpha. . 
Alsen . . 
Atlas . . 
Saylor's 



170,000 

324,000 

520,000 

45,000 



No. 
Briquettes. 



852 
1620 
2790 

243 



Neat 
Lbs. per Sq. In. 



7 Day. 28 Day. 



866 
700 
572 
744 
615 
453 
755 
643 
549 
815 
714 
669 



857 
716 
601 
864 

747 
650 

785 
662 

575 
883 

773 
694 



I c. 3 Ottawa Sand 
Lbs. per Sq. In. 



7 Day. 



320 

285 

246 

263 

192 

147 

324' 

279 

221 

265 
247 
208 



Day. 



458 
380 
336 
360 
300 
219 
414 
356 
279 
392 
354 
322 



The large quantities of cement used with the corre- 
sponding large number of briquettes tested give the Table 
special value and interest. The Ottawa sand is the 
standard silica sand of that name so extensively used in 
cement mortar testing. 

The preceding tabular results give ultimate tensile re- 
sistances for- periods no longer than 28 days, but both neat 
cement and cement mortars go on acquiring additional 
resistance for long periods, although at slow rates after a 
period of 28 days; indeed, it may be stated without ex- 
aggeration generally after a period of only 7 days. Table 
III therefore is used to show the increase of ultimate resist- 
ance up to a period of six months. The results of this table 
are taken from the Annual Report of the Bureau of Surveys 
of Philadelphia, Pa., for the year 1901. It will be observed 
that the values are not greatly different from those given 
in Table I at a date 10 years later. In fact, the earlier 
values are a little higher than the later, showing the ten- 
dency to secure a higher degree of permanency in the setting 
of the cement rather than higher ultimate tensile resistances. 



Art. 60.] CEMENT, CEMENT MORTARS, ETC.— BRICK. 



369 



Table III. 

AVERAGE RESULTS OF PORTLAND CEMENT TESTS MADE 
DURING 190L 



Brand. 



No. of 
Tests. 



Broken. 



Ultimate Tensile Resistance in Pounds per Square Inch. 



Neat . 



24 Hrs. 


7 Days. 


28 Dys. 


2 Mos. 


3 Mos. 


4 Mos. 


357 


770 


834 


885 


813 


785 


542 


728 


790 


802 


761 


815 


235 


336 


387 


443 






363 


826 


932 


778 






424 


669 


719 


713 


745 


776 


418 


830 


864 


775 






377 


699 


747 


684 


735 


774 


345 


721 


7-^3 








460 


800 


955 


775 






295 


697 


766 


756 


766 


733 


437 


721 


746 


731 


715 


727 


290 


748 


767 


707 


807 


710 


524 


713 


765 


788 


796 


775 



6 Mos. 



Alpha. . . . 
Atlas. . . . 
* Castle. . 
Dexter. . . 
Giant. . . . 
Krause's. 
Lehigh. . . 
Phoenix. . 
Reading.. 
Saylor's. . 
Star. . . . . 
Vulcanite. 
Whitehall 




827 
825 

786 
760 

745 
740 



Brand. 



No. of 
Tests. 



Broken. 



Ultimate Tensile Resistance in Pounds per Square Inch. 



to 3 Standard Quartz Sand. 



24 Hrs. 


7 Days. 


28 Dys. 


2 Mos. 


3 Mos. 


4 Mos. 


81 


252 


314 


344 


312 


302 


104 


204 


289 


324 


321 


337 


65 


121 


176 


215 






68 


298 


336 


312 






87 


227 


309 


328 


317 


328 


74 


229 


285 


270 






76 


233 


329 


296 


310 


303 


94 


264 


343 








150 


263 


301 


338 






64 


217 


296 


319 


301 


311 


77 


219 


298 


321 


301 


2S6 


45 


226 


287 


269 


298 


280 


87 


232 


313 


295 


295 


343 



6 Mos. 



Alpha. . .. 
Atlas. .. . 
* Castle. . 
Dexter. . . 
Giant. . . . 
Krause's. 
Lehigh. . . 
Phoenix. . 
Reading. . 
Saylor's. . 

Star 

Vulcanite. 
Whitehall 



262 
308 



329 

325 

286 
330 



370 



TENSION. 



[Ch. VII. 



During the construction of a number of dams in the 
Croton basin supplying the water works of the City of New 
York, briquettes of neat cement and of mortar i to 2 and 
I to 3 were tested after periods beginning with one week 
and extending up to five years. There was a continuous 
increase of ultimate resistance throughout the entire period, 
although at a very slow rate after about six months. At 
the end of five years the neat Portland cement attained 
an ultimate resistance of 840 pounds per square inch and 
the I to 2 mortar, 700 pounds per square inch, while the 
I to 3 mortar reached 590 pounds per square inch. 

Other tests of briquettes up to two years of age and 
more confirm the preceding results. 

The recent cement product, called silica-Portland 
cement, is manufactured by grinding together certain 
portions of clean silicious sand and Portland cement 
The results given below are taken from the tests of such 
silica-Pvortland cement, manufactured by the SiHca-Port- 
land Cement Co., of Long Island City, N. Y. One part, 
by weight, of Aalborg Portland cement was ground to- 



'Table IV.. 

SILICA-PORTLAND CEMENT. 

Ultimate Tensile Resistance in Pounds per Square Inch. 





Per Cent, 
of Water. 


Age. 


Mixture. 


Seven 
Days. 


Fifteen 
Days. 


Twentv-one 
Days. 


Two 
Hundred and 
Nineteen 
Days. ■ 


Neat 


18-21% 
11% 


( 148 

8 130 

( 121 

( ^' 
23 i 69 

( 58 


(172 

6-^ 165 

( 147 


( 166 
S\ 149 

( 121 
( 114 

8^ 98 
( 88 




(1-6) s. C.-2 q. .. 


( 220 

5] 204 
( 194 



All specimens one day in air and remainder in water. 



Art. 60.] CEMENT, CEMENT MORTARS, ETC.— BRICK. 371 

gether with six parts, by weight, of clean silicious sand 
to such a degree of fineness that essentially all of the 
product passed through a 32,000-mesh sieve. This finely 
ground mixture of i cement to 6 sand, by weight, is called 
"neat" in what follows, while " (1-6) s. c.-2 q." is i part, by 
weight, of the *'neat" silica-Portland cement to 2 parts, by 
weight, of crushed quartz, or ''standard" sand, all of whicli 
passes a No. 20 sieve and is retained on a No. 30 sieve. The 
results were obtained in the cement -testing laboratory of 
the department of civil engineering of Columbia Univei-sity. 
The figures on the left of the brackets show the number of 
tests of which the ultimate resistances are the greatest, 
mean, and least in each case. 

Five seven-day tests of the Aalborg Portland cement 
used in the manufacture of the silica-Portland cement 
gave the following greatest, mean, and least ultimate 
tensile resistances, the specimens having been one day in 
air and six days in water: 

Greatest. Mean. Least. 

594 lbs. per sq. in. 536 lbs. per sq. in. 441 lbs. per sq. in. 

Four specimens of the neat silica-Portland cement (1-6), 
one day in air and the remainder of the time in water, 
gave the following results: 

Age. 

308 lbs. per sq. in 199 days. ' 

264 " " 190 " 

294 " " 189 " 

L260 " " 185 " 



Neat (1-6). 



All the preceding tensile tests of cement and cement 
mortars, unless otherwise stated, were made with the shape 
of briquette shown in Fig. i, which was recommended for 
use in the report of the " Committee on a Uniform System 
for Tests of Cement " of the American Society of Civil 
Engineers. That report was made in 191 2, and the bri- 



37' 



TENSION. 



[Ch. vn. 



quette recommended has become the standard in American 
practice for the testing of cements and mortars. 




Fig. I. 



Weight of Concrete. 

As concrete is frequently used in masses where weight 
is an important element, it is always desirable to use an 
aggregate of high specific gravity. Concrete w^hen made of 
cement, sand and silicious gravel or broken limestone, trap- 
rock or granitic rock in such mixtures as are commonly 
employed, will weigh from 140 to 155 pounds per cubic foot 
with the greater part running from 145 to 150 pounds per 
cubic foot. 

The weight of cinder concrete will necessarily vary much 
with the character of the cinders. It may usually be taken 
as weighing about two-thirds as much as ordinary concrete 



Art. 60. CEMENT, CEMENT MORTARS, ETC.— BRICK. 373 

made with gravel or broken stone, i.e., from 100 to no 
pounds per cubic foot. 

Adhesion between Bricks and Cement Mortar. 

General Q. A. Gillmore many years ago investigated 
the adhesion of bricks to the cement mortar joint between 
them and also the adhesion of fine-cut granite to a similar 
joint. As might be expected in connection with such tests 
his results varied greatly, the highest belonging to a rich 
cement mortar and the lowest to the lean mortar of i 
cement to 6 sand. He found the adhesion to vary from 
about 31 pounds per square inch for neat cement to brick 
to nearly 3.3 pounds per square inch for a lean mortar of 
I cement to 6 sand. With fine-cut granite the adhesion 
for neat cement was 27.5 pounds per square inch and for 
cement mortar of i cement to 4 sand about 8 pounds per 
square inch. It is highly probable that the actual adhesion 
of bricks and cut stone to the usual joints made of i cement 
to 2 sand or i cement to 3 sand would be materially less 
in a mass of masonry than as arranged for a laboratory 
test. Nevertheless these early investigations would indi- 
cate that such joints might be worth from 8 to 12 pounds 
per square inch for bricks and but little different for 
granite. 

Mr. Emil Kuichling prepared a paper in 1888 from all 
available sources for the purpose of disclosing what all 
experimental investigation had determined up to that time. 
These results indicated that neat cement might give ad- 
hesion to bricks or cut stone varying from about 20 pounds 
up to over 200 pounds per square inch, with values from 
29. pounds up to 146 pounds per square inch for mortar of 
I cement to i sand; and 38 pounds to 73 pounds per square 
inch for a mortar of i cement to 2 sand. Further, accord- 
ing to his table a mortar of i cement ^to 3 sand would 



374 



TENSION. 



[Ch. VII. 



yield adhesion from 13 pounds up to 48 pounds per square 
inch and but httle less for a mortar of i cement to 4 sand. 
Nearly all these results, however, are undoubtedly too high 
for the usual masses of masonry in engineering construction. 

Other experimental determinations of the adhesive 
resistance of natural and Portland cement mortars to 
brick and stone may be found in the report of the Chief of 
Engineers, U. S. A., for 1895. At the age of 28 days 
the adhesive resistance of neat Portland cement to the 
surface of sawn limestone was about 270 poimds per square 
inch; about 240 pounds per square inch with a mortar of 
I cement to J sand; about 225 pounds per square inch 
with a mortar of i cement to i sand, and about 170 pounds 
per square inch with a mortar of i cement to 2 sand. 

Table V exhibits the average results of three and six 
months' tests of the adhesion of Portland and natural 
cement mortars to bricks w^hich were cemented to each 
other at right angles and then pulled apart normally at the 
ends of the periods named. These average results are 
taken from the same report of the Chief of Engineers, 
U. S. A., for 1895- 

Table V. 

AVERAGE ADHESIVE RESISTANCE OF BRICKS CEMENTED 
TOGETHER AT RIGHT ANGLES TO EACH OTHER. 



Cement. 


Mortar. 


Adhesion, 
Pounds per Square Inch. 


Portland 


Neat 


60 


* ' 


I c, i s. 


60 


< < 


I C, I s. 


40 


( ( 


I c, 2 s. 


20 


(( 


I c, 3 s. 


20 


Natural 


Neat 


55 


( ( 


I c, ^ s. 


50 


< ( 


I C., I s. 


45- 


( < 


I C., 2 S. 


30 


(< 


I c., 3 s. 


15 



Art. 60.] CEMENT, CEMENT MORTARS, ETC.— BRICK. 375 

There will also be found in that report average values 
of the shearing adhesion of plain i-inch round bolts to neat 
Portland cement and to Portland cement mortars of i 
month's age, the bolts having been embedded at various 
depths "from 2 to 10 inches in the mortars. The shearing 
adhesion for the neat cement varied from a maximum 
of 345 potmds per square inch for a depth of insertion of 4 
inches down to 230 pounds per square inch for a depth of 
insertion of about 8J inches. In the case of the Portland 
cement mortar of i cement to 2 sand the shearing adhesion 
varied from a maximum of 280 pounds per square inch for a 
depth of insertion of the bolt of 2 J inches down to 250 
pounds per square inch for a depth of insertion of about 
7 1 inches. When the bolt was embedded in the Portland 
cement mortar of i cement to 4 sand the shearing adhesion 
ranged from a maximum of about 145 poimds per square 
inch for a depth of insertion of 10 inches to a minimum of 
about 70 pounds per square inch for a depth of insertion 
of 2 inches. These values of shearing adhesion are impor- 
tant results in the theory and design of concrete-steel 
members. 

The Effect of Freezing Cements and Cement Mortars. 

There have been many attempts made to determine the 
effect of freezing neat cements and cement mortars after 
having been mixed for use at various ages and under 
various conditions. vSome valuable data have been ac- 
cumulated, but the conditions attending such investiga- 
tions are so complicated and so difficult to be analyzed 
quantitatively that many most discordant conclusions have 
been reached. Different results will follow if the freezing- 
is done immediately after the mixing of the cement or 
mortar, or after the initial set has taken place, or after the 
considerable hardening which takes place at the age of 



376 TENSION. [Ch. VII. 

12 to 24 hours. Probably the best data in this connection 
arise from an engineer's practical experience in laying 
masonry when the temperature of the air is below the 
freezing-point. Under such circumstances it is rarely 
the case that anything more than surface freezing takes 
place before the hardening of Portland cement. With the 
slower action of the natural cements similar conditions do 
not exist. It is undoubtedly prejudicial even with Port- 
land cements to have alternate freezing and thawing take 
place at comparatively short intervals of time. On the 
other hand, the great majority of laboratory investigations 
indicate that Portland cement or cement mortars may be 
severely frozen and remain so for long periods of time 
without essential injury. It is probable that setting usually 
proceeds during a frozen condition, but at an exceedingly 
slow rate, and that the operation of setting is actively 
renewed after thawing. 

While it has been stated in some quarters that natural 
cements may be frozen similarly and thawed without 
essential injury, there is considerable laboratory evidence 
as well as that of practice which indicates that conclusion 
to be erroneous, especially if it be given any considerable 
application. There may be cases in which natural cements 
can be or have been frozen without essential injury, but 
the author's experience in extended practical operations in 
masonry construction induces him to believe that any 
natural cement severely frozen before being thoroughly 
hardened is so seriously injured as to be practically de- 
stroyed. On the other hand, his extended observations 
not only on his own work, but on those of others, lead him 
to beheve that, as a rule, Portland cement will not be 
sensibly injured under the conditions of actual masonry 
construction by being frozen. It is customary in most 
large works to permit no masonry to be laid at a tempera- 



Art. 60.] CEMENT, CEMENT MORTARS, ETC.— BRICK, 377 

ture much below about 26° Fahr. above zero, but with 
precautions easily attained it is certain that concrete and 
other masonry laid in Portland cement mortar may prop- 
erly and safely be put in place several degrees below that 
temperature. 

It has also been stated in some quarters that natural 
cements and some Portlands have been actually improved 
by being frozen. Such conclusions should be received with 
exceeding caution. The author believes that there is no 
conclusive evidence that any cement or cement mortar can 
be improved by freezing. 

In cold weather it is customary on some works to use 
salt water for mixing mortars and concretes, and that 
practice when suitably conducted may be resorted to with 
safety and propriety. Such solutions generally rtm from 
2 to 8 or 10 per cent, by weight of salt. Occasionally, also, 
soda is dissolved in water at tlie rate of 2 pounds per gal- 
lon. Before using this solution an equal volume of water 
is added so that the final solution contains about i pound 
of soda to a gallon of water. This solution expedites 
the setting of the cement with a view to accomplishing 
a safe degree of hardening before the mortar is frozen. 
It is doubtful whether this practice should be encouraged. 

The Linear Thermal Expansion and Contraction of 
Concrete and Stone. 

Satisfactory investigations regarding the expansion and 
contraction of concrete and stone are exceedingly few in 
number, and the data by which variations in the dimen- 
sions of large masses of masonry due to temperature changes 
can be computed are correspondingly meagre. Professor 
William D. Pence, of Purdue University, has made such 
investigations and presented the "results in a valuable 
paper read before the Western Society of Engineers, 



37^ 



TENSION. 



tCh. VII. 



November 20, 1901. In his experimental work he com- 
pared the thermal linear changes of concrete bars and bars 
of steel and copper, basing the coefficients of expansion of 
the concrete and mortar on the relative changes of the 
two materials for the same range of temperature. These 
experiments w^ere conducted with great care, but the 
resulting values might perhaps have been at least better 
defined had two materials been employed with a greater 
difference in their rates of thermal expansion and contrac- 
tion. Professor Pence employed two kinds of concrete and 
one bar of Kankakee limestone, seven experiments having 
been performed on a concrete of i Portland cement, 2 sand, 
and 4 broken stone ; one on a concrete of i Portland cement, 
2 sand, and 4 gravel ; and three on a concrete composed of i 
cement and 5 of sand and gravel, making the mixture 
essentially equivalent to the preceding concrete of i cement, 
2 sand, and 4 gravel. The maximum, mean, and minimum 
coefficients of linear expansion per degree Fahr. found in 
these tests were as follows: 



Kind of Concrete. 


Maximum. 


Mean. 


Minimum. 


Broken stone, 1:2:4 

Gravel, 1:2:4 

Gravel 1*5 


.0000057 
.0000055 


.0000055 
.0000054 
.0000053 
.0000056 


.0000052 
.0000052 


Kankakee limestone 







Betw^een January and June, 1902, Messrs. J. G. Rae 
and R. E. Dougherty, graduating students in Civil Engineer- 
ing at Columbia University, with the aid of Professor 
Hallock of the Department of Physics of the same university, 
determined with great care by the most accurate direct 
measurements the coefficients of linear thermal expansion 
of one bar of concrete of i Portland cement, 3 sand and 5 
gravel, and one bar of mortar of i Portland cement and 2 
sand, each bar being 4 inches by 4 inches in cross-section 



Art. 6 1.] TIMBER IN TENSION. 379 

and about 3 feet long, both bars being tested at the age 
of about 5i years. The coefficients of linear thermal 
expansion for each degree Fahr. found in these investiga- 
tions were as follows: 

For 1:3:5 concrete 00000655 

" 1:2 mortar 00000561 

It is believed that these last two determinations were 
made with the utmost accuracy attainable at the present 
time in an unusually w^ell equipped physical laboratory 
and under most favorable conditions. 

When it is remembered that the coefficient of linear 
thermal expansion of such iron and steel as are used in 
engineering structures is about .0000066,* it is apparent 
that structures of combined concrete or other masonry and 
steel may be expected to act under thermal changes essen- 
tially as a unit, a conclusion which is justified at the present 
time by extended experience. 

Art. 61. — Timber in Tension. 

The ultimate resistance of timber in general is much 
affected by the moisture which it contains, except that the 
amoimt of moisture does not appear to affect sensibly the 
ultimate tensile resistance. At this point, therefore, no 
further attention will be given to the effect of moisture or sap 
on the tensile resistance, but the infliience of moisture on the 

* A large number of determinations of the thermal expansion of iron and 
steel per degree Fahr. may be found in the U. S. Report of Tests of Metals 
and Other Materials for 1887, The maximum, mean, and minimum for 
steel bars are as follows: 

. 000006 756 . 000006 466 . 000006 1 7 

Other coefficients of thermal expansion are also given as follows: 

Wrought iron 00000673 

Cast iron 000005926 

Copper 000009 1 29 



38o 



TENSION. 



[Ch. VII. 



compressive and bending resistances will be fully set forth 
in the articles devoted to timber in compression and bending. 

There are few results of investigations which give satis- 
factory moduli of elasticity for timber in. tension. Values 
are given in the annual " U. S. Report of Tests of Metals 
and Other Materials," but these results are generally for 
small selected sticks which are quite different from com- 
mercial sizes of lumber as generally used. Some of these 
moduli run up to nearly 3,000,000, which is much too high for 
any ordinary commercial timber as used in structural work. 

In " Tests of Structural Timbers," by McGarvey Cline, 
Director of Forest Products Laboratory, and A. L. Heim, 
Engineer of Forest Products, issued as Bulletin 108 of the 
U. S. Department of Agriculture, 191 2, a large number of 
determinations are made of ultimate resistance, elastic limit 
and modulus of elasticity for commercial sizes of lumber 
of nine different kinds of generally used timber. The 
moduli of elasticity, however, are determined from bending 
tests, which makes them a kind of composite of both tension 
and compression values. The results found, however, are 
among the best available. 

The following tabular statement gives the moduli for 
green and air-seasoned structural sizes: 

Table I. 



Green 



Air-seasoned 



Wt. per 

cu. ft. 

oven-dry 



Long-leaf Pine . . 

Douglas Fir 

Short-leaf Pine . . 
Western Larch . . . 
Loblolly Pine . . . . 

Tamarack 

Western Hemlock 

Red Wood 

Norway Pine . . . . 



1,463,000 
1,517,000 
1,473,000 
1,301,000 
1,387,000 
1,220,000 
1,445,000 
1,042,000 
1,133,000 



1,705,000 
1,549,000 
1,726,000 
1,487,000 
1,487,000 
1,341,000 
1,737,000 
890,000 
1,418,000 



35 
28 

30 
28 
31 
30 

27 
22 
25 



Art. 6i.] TIMBER IN TENSION. 381 

It will be noticed that redwood gives the lowest 
modulus of elasticity and Norwa}^ pine next above it 
except the value for air-seasoned tamarack. Long-leaf 
pine, short-leaf pine, and Douglas fir give nearly the 
same results. 

In determinmg the tensile resistance, and, indeed, other 
resistances of timber, the size of the specimen plays a 
more important part, probably, than in any other class 
of materials used by the engineer. Small specimens, such 
as are usually employed in tensile tests, are inevitably so 
selected as to eliminate such defects as decay and decayed 
or other , knots, wind shakes, season cracks, and other 
deteriorating features, so that the results exhibit physical 
properties belonging to the best parts of full-size sticks. 
In engineering practice, on the other hand, large pieces of 
timber must be used as furnished in the timber market. 
However close the inspection, ma}^ be such pieces in- 
variably include within their volumes m^any spots of weak- 
ness due to those features which in the small specimen are 
carefully excluded. It is of the utmost consequence, 
therefore, in dealing with physical data belonging to timber 
to realize that results determined by the testing of small 
specimens are almost without exception materially mis- 
leading in consequence of reaching higher values than those 
which can possibly belong to the average stick used in 
structural work. These observations must be carefully 
remembered in considering the experimental data which 
follow. 

While there exists a large amount of data on the tensile 
tests of timber it relates largely to small selected sticks 
or is otherwise scarcely available for engineering construc- 
tion. The best recent data are given by Messrs. Cline and 
Heim from which Table I was taken. On page 57 of that 
Bulletin tabulated data of a large number of bending tests 



382 



TENSION. 



[Ch. VII. 



of green and dry structural timbers are found, the failures 
being by tension in the fibres subjected to that kind of 
stress. Those data are shown in Table II. The modulus 
of rupture is simply the intensity of stress in the most 
remote fibre of the timber. 

Table IL 



Species. 



Long-leaf pine: 

Green 

Dry 

Douglas fir: 

Green 

Dry 

Short-leaf pine: 

Green 

Dry 

Western larch: 

Green 

Dry . . 

Loblolly pine: 

Green 

Dry 

Tamarack : 

Green 

Dry 

Western hemlock 

Green 

Dry 

Redwood : 

Green 

Dry. . 

Norway pine: 

Green 

Dry 



Average 

modulus 
of rupture All 

lbs. per j tension 
failures. 



Modulus of rupture in per cent, of average green 
modulus of rupture. First failure by tension (per cent.) 



6,140 

5.749 

5.983 
6,372 

5.548 
6,573 

4.948 
5.856 

5.084 
6,118 

4.556 
5,498 

5.296 
6,420 

4.472 
3,891 

3,864 
6,054 



112 
121 

83 

82 

94 
117 

73 
no 

86 
120 

90 
106 

74 
108 

81 
80 

94 
134 



Failure due to 



Large 
knots. 



82 
69 

76 
96 

166 
102 

73 
39 

71 



71 

77 



94 
136 



Small 
knots. 



80 

78 

90 



77 
112 

86 



90 



92 

58' 



Irregu- 
lar grs. 



77 
82 



100 
115 

71 
103 

90 
114 

96 
112 



106 

55 
48 

73 



Pitch 
pockets 



90 



Nothing 
apparent 



112 
121 

104 
136 

109 
132 

100 
124 

98 
138 

98 
100 

81 
119 

90 

87, 

105 
129 



The moduH of rupture are the averages of all failures 
whether by tension, compression or shear, but the figures 
given in the table after the second column represent the 



Art. 6i.] 



TIMBER IN TENSION. 



383 



percentages of the average '* green " moduli of rupture at 
which the extreme fibres failed in tension under influence 
of " large knots," " small knots," *' irregular grain " or 
" nothing apparent " as indicated at the head of each 
column. Although these values are not found by direct 
tests of tension, they may be accepted as fair and suitable 
ultimate resistances of the different kinds of timber in 
tension. 

Table III. 



Kind of Timber. 



Ultimate Resistance, 

Pounds per 

Square Inch. 



With 
Grain. 



Across 
Grain. 



Working Stresses, 
Pounds per 
Square Inch, 



With 
Grain. 



Across 
Grain . 



White oak 

White pine . 

Southern long-leaf or Georgia yellow 

pine 

Douglas, Oregon, and yellow fir 

Washington fir or pine (red fir) 

Northern or short -leaf yellow pine. . . . 

Red pine 

Norway pine 

Canadian (Ottowa) white pine 

Canadian (Ontario) red pine 

Spruce and Eastern fir 

Hemlock 

Cypress 

Cedar 

Chestnut 

California redwood 

Cahfomia spruce 



10,000 
7,000 

12,000 

12,000 

10,000 

9,000 

9,000 

8,000 

10,000 

10,000 

8,000 

6,000 

6,000 

8,000 

9,000 

7,000 



2,000 
500 

600 



500 
500 



500 



1,000 
700 

1,200 

1,200 

1,000 

900 

900 

800 

1,000 

1,000 

800 

600 

600 

800 

900 

700 



200 
50 

60 



50 
50 



50 



Reviewing all the experimental work which has been 
done up to the present time (1902) in determining the 
ultimate tensile resistance of timber, and keeping in view 
experience with the resistance of full-size timber sticks 
in completed structures, the best representative series of 
values of the ultimate and working tensile intensities of 
timbers is that recommended by the Committee on 



384 TENSION. (Ch. VII. 

" Strength of Bridge and Trestle Timbers " of the Associa- 
tion of Railway Superintendents of Bridges and Buildings 
at the Fifth Annual Convention in New Orleans, 1895. 
That series is given in Table III. 

The ultimate resistances of the table are much too 
high for full size pieces, but the working stresses may be 
accepted as they stand. 

It will be noticed that the ultimate tensile resistance 
of the various timbers across the grain, so far as they are 
given, are but small fractions of the ultimate resistances 
along the grain. A corresponding large decrease in resist- 
ance across the grain will also be found in connection with 
the compressive resistance of the same timbers. The 
working resistances given in this table are those employed 
in the great bulk of engineering timber structures. 



CHAPTER VIII. 

COMPRESSION. 

Art. 62. — Preliminary. 

With the exception of material in the shape of long 
columns, but few experiments, comparatively speaking, 
have been made upon the compressive resistance of con- 
structive materials. 

Pieces of material subjected to compression are divided 
into two general classes — " short blocks " and ** long col- 
umns "; the first of these, only, afford phenomena of pure 
compression. 

A " short block " is such a piece of material that if it be 
subjected to compressive load it will fail by pure compres- 
sion. 

On the other hand, a long column (as has been indi- 
cated in Art. 35) fails by combined compression and bending. 

Short blocks only will be considered in the articles 
immediately succeeding, while long columns will be sepa- 
rately considered further on. 

The length of a short block is usually about three times 
its least lateral dimension or less. 

It has already been shown in Art. 5 that the greatest 
shear in a short block subjected to compression will be 
found in planes making an angle of 45° with the surfaces 
of the block on which the compressive force acts, i.e., with 

385 



386 COMPRESSION. [Ch. VIII 

its ends. If the material is not ductile this shear will 
frequently cause wedge-shaped portions to separate from 
the block. But the friction at these end surfaces, and in 
the surfaces of failure will 'prevent those wedge portions 
shearing off at^ that angle. In fact the friction will cause 
the angle of separation to be considerably larger than 45°; 
let it be called a. Then, in order that there maybe perfect 
freedom in failure, the length of the block must not be less 
than its least width or breadth multiplied by 2 tan a. In 
some cases, a has been found to be about 55°, for which 
value. 

2 tan a = 2 X 1.43 = 2.86. 

If the bearing faces of the short block under compres- 
sion are of much area, for such a purpose, it will be difficult 
in many cases, especially with large loads, to secure a 
uniform apph cation of those loads. The resulting ultimate 
resistance for the entire block will give an average intensity 
of pressure which may be quite different from the greatest 
intensity. These simple considerations are particularly 
pertinent to such materials as blocks of concrete or of 
natural stone, which may be 12 inches square or more in 
section. 

Again, in such material as natural or artificial stone the 
friction between the head of the testing machine and the 
bearing surface of the specimen, or along the planes of 
greatest ultimate shear will tend to support laterally to 
some extent the material as it approaches failure, thus 
raising the apparent ultimate resistance of the material. 
The shorter the block the greater will be this frictional 
supporting tendency. This effect has been marked where 
the tests specimens have been cubes varying from 2 inches 
on their edges to 12 inches, the large cubes showing mate- 
rially greater resisting capacity. 




Failure of short cylinders of cast iron showing the 
shearing of the metal on the plane of maximum 
shear. 




View exhibiting the failure of short cylinders of Connecticut brown sandstone. 

{To face page 386.) 



Art. 63.] WROUGHT IRON. 3S7 

Art. 63. — Wrought Iron. 

It is difficult to fix the point of failure of a short block 
of wrought iron or other ductile material. As the load 
increases above the elastic limit, the cross-sections of the 
test piece increase in lateral dimensions or *' bulge out," 
so that increase of compressive force simply produces an 
increased area of resistance, while the material never truly 
fails by crumbling or shearing off in wedges. 

It is comparatively easy to determine the elastic limit, 
but at what degree of loading the material may be said to 
fail after permanent distortion begins is not clear unless 
some arbitrary limit should be fixed by convention. 

In an actual structure obviously failure may be said 
to take place when the degree of distortion is such that the 
structure fails to discharge safely its function as a load 
carrier, but that degree of distortion would vary much in 
different structures or in different parts, possibly, of the 
same structure. 

For the present purpose it may perhaps be assumed 
tentatively that a ductile material fails when its distortion 
under compressive loading becomes apparent to the unaided 
eye. 

Modulus of Elasticity. 

As wrought iron is no longer a structural material, there 
are practically no recent tests to determine the compressive 
modulus of elasticity, but earlier investigators made suf- 
ficient tests when the material was in general use to establish 
the modulus with reasonable accuracy. Those investi- 
gations show that there is no essential difference between 
moduli for compression and tension. Hence the modulus 
of elasticity for wrought iron in compression may be taken 
at 26,000,000. Small specimens would in some cases yield 



388 COMPRESSION. (Ch. VIII 

results perhaps as high as 28,000,000, but for general use 
the former or smaller value is preferable. 

Limit of Elasticity and Ultimate Resistance. 

Investigations for determining the elastic limit of 
wrought iron in compression are almost entirely lacking, 
but its value may safely be taken the same as for tension, 
i.e., depending upon the area of cross-section and the 
amount of work put upon the material in its manufacture, 
from 22,000 to perhaps 26,000 pounds per square inch, the 
former for large sections and the latter for small sections. 
The difficulties met in the effort to determine a well-defined 
ultimate compressive resistance for wrourht iron have 
already been noticed, but such compression tests as were 
made during the general use of wrought iron for structural 
purposes indicate that what may be termed the ultimate 
compressive resistance may reasonably be taken at about 
the ultimate tensile resistance. The amount of permanent 
distortion taking place at that degree of loading has not 
been satisfactorily determined, but it would certainly be 
apparent to the unaided eye and it might run from i per 
cent, to 5 per cent, or possibly more. It m,ay be assumed, 
therefore, thset the ultimate compressive resistance of 
wrought iron will range generally from 45,000 to 50,000 
pounds per square inch. 

Art. 64. — Cast Iron. 

The behavior of cast iron under compression as found 
in ordinary casting is not less erratic than in tension. When 
this material was used for such purposes as heavy ordnance 
and car wheels it was so produced as to possess excellent 
physical qualities for a cast metal, especially after remiclting 
and being held in fusion. Even then, however, the modulus 



Art. 64.] CAST (RON. 389 

of elasticity was not much higher than for the best quaUties 
of ordinary castings. It may be said generally that the 
modulus of elasticity for cast iron in either tension or com- 
pression may be taken from 12,000,000 to 14,000,000. 
These values are about half of the corresponding values for 
wrought iron and little less than half the corresponding 
values for structural steel. 

Inasmuch as cast iron is a brittle material failing 
suddenly at the limit of its resisting capacity, either in 
tension or compression, it can scarcely be said to have an 
elastic limit except for special grades of unusual excellence, 
and even with such material it is not well defined. 

The ultimate resistance of cast iron to compression is 
fairly well defined, but it varies greatly in value according 
to its quality. Special grades for ordnance and car wheels 
may have compressive resistances running from 100,000 
per square inch up to 150,000 pounds per square inch. 
For many years when cast-iron columns were used in engi- 
neering practice it was customary to consider the ultimate 
compressive resistance for such members as 100,000 pounds 
per square inch, but that value is far too high. Although 
the quality of ordinary castings is variable, it is reasonable 
to take the ultimate compressive resistance at 80,000 pounds 
per square inch for such material as may be used under 
good and effective specifications for columns, machine 
frames and similar purposes, although there are modern 
cast-iron column tests which appear to indicate that even 
that value is too high. 

Art. 65.— Steel. 

Table I of Art. 58 contains the results found by Prof. 
Ricketts in testing cylindrical specimens of mild steel in 
compression. These specimens were six inches long be- 
tween carefully faced ends, and, as the table shows, their 



39° COMPRESSION. [Ch. VIII. 

diameter was about 0.75 inch. The coefficients of 
elasticity for compression were found by measurements 
very carefully made with a micrometer on a length of four 
inches. The elastic limits, however, were determined by 
operating with a cylinder two inches long, and were taken 
at those points where the material of the specimens ceased 
to hold up the scale beam, and may have been somewhat 
above that point where the ratio between stress and strain 
ceases to be essentially constant. 

The coefficients of elasticity are found to be quite 
uniform, irrespective of the per cents of carbon, within the 
limits of the table, and they are seen to be a very little 
less than the coefficients for tension. The difference is 
so small that no essential error will arise if, for all en- 
gineering purposes, they are assumed the same. 

A comparison of the elastic limits for tension and 
compression presents some irregularities; yet with the 
exception of the high percentages of carbon in the last two 
grades of Bessemer metal, the two sets of elastic limits as 
wholes are not very different from each other. In the 
Bessemer steel with the two high per cents of carbon, the 
tensile elastic limits are materially higher than those for 
compression. The following very important conclusion 
results from this comparison of the elastic limits for the 
mild structural steels: since these elastic limits are es- 
sentially equal it is not only permissible but wholly rational 
to increase the working resistances of mild steel bridge 
columns over tjiose for iron in at least the same proportion, 
that the tensile working stress of the same steel is increased 
over that of iron in tension. Experiments on a sufficient 
number of full-size steel columns are yet lacking to verify 
this conclusion. 

It appears from such data on the compressive resistance 
of steel as exist that not only the coefficient of elasticity 



Art. 65.] STEEL, 391 

but, also, the limit of elasticity in compression may be 
taken the same as that for tension for the same grade of 
steel. This was practically true in the older investiga- 
tions of Kirkaldy, and it is essentially confirmed in the 
few later investigations available. 

The ultimate compressive resistance of steel, like the 
ultimate tensile resistance, varies with the content of 
carbon, being comparatively low with a small percentage 
of carbon, and correspondingly large with a high percentage 
of that element. It is also much affected by the operations 
of tempering and annealing. 

Special grades of steel adapted to heat treatment have 
after such treatment given ultimate compressive resistances 
of various values up to nearly or quite 400,000 pounds per 
square inch and values ranging from 150,000 pounds up 
to 300,000 pounds per square inch are not uncommon in 
the records of the older testing. Such high results, however, 
are only obtained with hardened and tempered metal. 

There is the same uncertainty as to the point at which 
compressive failure takes place in steel which attaches to 
the ultimate compressive resistance of all ductile metals 
and which was commented upon in Art. 63. It is probably 
safe, however, if not entirely correct, to take the ultimate 
compressive resistances of different grades of steel equal 
to their ultimate tensile resistances in the absence . of 
explicit determinations; and a similar observation may 
be applied to the working resistances in pure compression 
of same grades of steel. 

Art. 66. — Copper, Tin, Zinc, Lead, and Alloys.* 

Table I shows some coefficients of elasticity (i.e., ratios 
between stress and strain), computed from data deter- 

* As this field of investigation has not been worked since Prof. Thurston 
left it his results are ^llowed to stand (19 15). 



392 



COMPRESSION. 



[Ch. VIII. 



mined by Prof. Thurston, and given by him in the " Trans. 
Amer. Soc. of Civ. Engrs.," Sept., 1881. The gun bronze 
contained copper, 89.97; "tin, 10.00; flux, 0.03. The cast 
copper was cast very hot. 

Table I. 



Stress in Pounds 
per Square Inch. 


Coefficients of Elasticity in Pounds per Square Inch; 


Gun Bronze. 


Cast Copper. 


1,620 

3,260 

6,520 

9,780 . 
13,040 
16,300 
19,560 
22,820 
26,080 
29,340 
32,600 
48,900 


3,622,000 
4,075,000 
6,113,000 
6,520,000 
5,433,000 
5,148,000 
3,935,000 
2,308,000 

1,073,000 

463,600 


1,254,000 
1,415,000 
1,651,000 
1,795,000 
1,824,000 
1,842,000 
1,845,000 
1,735,000 
1,503,000 
1,144,000 
815,000 
332,500 



The ratios of stress over strain are far from being con- 
stant. Strictly speaking, therefore, there is no elastic 
limit in either case. In that of the gim bronze, however, 
it may be approximately taken at 20,000 pounds per square 
inch (Prof. Thurston takes it 22,820), and in that of the 
copper at 25,000 pounds. The test specimens v/ere two 
inches long and turned to 0.625 inch in diameter. 

At 38,000 pounds per square inch the gim bronze speci- 
men was shortened about 41 per cent, of its original length, 
while its diameter had become 0.77 inch. 

The copper specimen failed at 71,700 pounds per square 
inch, having been shortened about one third of its length. 

The results of a series of tests by Prof. Thurston, in 
connection with the United States testing commission, are 
given in Table II; they were abstracted from " ]\Iechanical 



Art. 66.] COPPER, TIN, ZINC, LEAD AND ALLOYS. 393 

Table II. 



Composition. 


Pounds per Square Inch 
Causing a Shortening of 


Greatest 
Load in 


Per Cent, 
of Short- 


Ultimate 
Crushing 














Pounds 


ening 


Resist- 


Manner of 














Caused 


ance in 


Failure. 


Copper. 


Tin. 


5 Per 

Cent. 


10 Per 
Cent. 


20 Per 

Cent. 


per 
Square 
Inch. 


by 

Greatest 

Load. 


Lbs. per 
Sauare 
Inch. 


07.83 


1.92 


29,340 


34,000 


46,000 


46,260 


0.37 


34,000 


Flattened 


95-96 


3-80 


39,200 


42,050 


52,150 


52,150 


0.30 


42,050 


" 


92.07 


7-76 


31,500 


42;000 


65,000 


84,100 


0.45 


42,000 


" 


90.43 


9-50 


32,000 


38,000 


60,000 


61,930 


0.34 


38,000 


" 


87.15 


12.77 


39,000 


53,000 


80,000 


89,640 


0.39 


53,000 


" 


80.99 


18.92 


65,000 


78,000 


103,490 


103,490 


. 20 


78,000 


" 


76.60 


23.23 


101,040 






114,080 


0.09 


114,080 


Crushed 


69.90 


29.85 








146,680 


0.04 


146,680 


" 


65-31 


34.47 








84,750 


. 03 


84,750 


" 


61.83 


37.74 








39,110 


0.02 


39,110 


" 


47-72 


51-09 










84,750 


0. 02 


84,750 


" 


44.62 


55-15 








35,850 


O.OI 


35,850 


" 


38.83 


60.79 









39,110 


0.02 


39,110 


" 


38.37 


61.32 








29,340 


O.OI 


29,340 


" 


34-22 


65.80 


19,560 






19,560 


0.06 


19,560 


" 


25.12 


74-51 


17,930 


17,930 


17,030 


17,930 


0.28 


17,930 


" 


20.21 


79-62 


16,300 


16,300 


16,300 


16,300 


0. 29 


16,300 


" 


15-12 


li-^^ 


6,520 


6,520 


6,520 


9,450 


0.51 


6,520 


Flattened 


11.48 


88.50 


10,100 


10,100 


10,100 


14,020 


0. 50 


10,100 


" 


8.57 


91.39 


6,500 






9,780 


0.06 


9,780 


" 


3.72 


96. 31 


6.520 


6,520 


6,520 


9,780 


0.34 


9,780 


" 


0.74 


99-02 


6,520 


6,520 


6,520 


9,780 


0.36 


9,780 


" 


°-32 


99-46 


6,520 


6,520 


6,520 


9,780 


0.38 


9,780 


« 


Cast c 


opper 


26,000 


39,000 


51,000 


74,970 


0.45 


39,000 


"■' 


" . 


" 


33,000 


45,500 


58,670 


78,230 


0.43 


45,500 


** 




II 


34,000 


42,000 


58,000 


71,710 


0.32 


42,000 


II 




" 


30,000 


36,000 


50,000 


104,300 


0.52 


36,000 


" 


" 


" 


30,000 


37,000 


50,000 


91,270 


0.48 


37,000 


iS 


*' 


" 


35,000 


48,000 


65,000 


97,790 


0.41 


48,000 


" 


Cast 


tin 


6,030 


6,400 


6,530 


7,500 


0.44 


6,400 





and Physical Properties of the Copper- tin Alloys," United 
States Report, edited by Prof. R. H. Thurston, 1879. All 
the specimens were 0.625 i^ch in diameter and 2 inches 
long. Scarcely one of them can be said to possess an 
elastic limit. 

The series of alloys presents some interesting results. 
About the middle third of the series are seen to be brittle 
compounds giving (as a rule) high ultimate compressive 
resistances, while the other two thirds are ductile, and give 
at the copper end high r-esults, and low ones at the tin end. 

It will be observed that Prof. Thurston took the load per 
square inch which gave a shortening of 10 per cent, of the 
original length as the. ultimate resistance to crushing of the 



394 



COMPRESSION. 



[Ch. VIII. 



ductile alloys and metals, since such materials cannot be 
said to completely fail under any pressure, but spread 
laterally and offer increased resistance. 



Table III. 



Per Cent, of 


Pounds per Square Inch for 














Per Cent, of 


Manner of 










Copper. 


Zinc. 


El. 


Ultimate 
Resistance. 


Shortening. 


Failure, 


96.07 


3-79 


305,500 


29,000 


:>.o 


Flattened 


90.56 


9.42 


342,100 


30,000 


10. 


* * 


89.80 


10.06 




29,500 


10. 


It 


76.65 


23.08 


656,500 


42,000 


10. 


tt 


60.94 


38-65 


1,772,500 


75,000 


10. 


tt 


55-15 


44-44 




78,000 


10. 


it 


49.66 


50.14 


1,345,500 


117,400 


10. 


<( 


47-56 


52.28 


1,500,000 


121,000 


10. 


- 


25.77 


73-45 


4,232,800 


110,822 


5-85 


Crushed 


20.81 


77-63 


2,485,000 


52,152 


2.75 


" 


14.19 


85.10 


897,000 


48,892 


10.8 


i ( 


10.30 


88.88 




49,000 


10. 


Flattened 


4-35 


94.59 




48,000 


10. 


** 


0.00 


100.00 


318,500 


22,000 


10. 





Table III contains the results of Prof. Thurston's tests 
of the copper-zinc alloys made while he was a member of 
the United States Board. The data are taken from ''Ex. 
Doc. 23, House of Representatives, 46th Congress, 2d 
Session." The specimens were two inches long and 0.625 
inch in diameter of circular cross-section. 

The values of E^ (ratios of stress over strain) are com- 
puted for about one quarter the ultimate resistance. This 
ratio is so very variable for different intensities of stress, 
that these alloys can scarcely be said to have a proper 
"elastic limit." 

Two specimens of tobin bronze, each .75 inch in diameter 
and I inch long, tested by the Fairbanks Company of New 
York City in 1891, were compressed about .8 per cent, at 
45,000 pounds per square inch, and a little over 10 per cent. 



Art. 67.] CEMENT— CEMENT MORTAR— CONCRETE. 395 

at 90,000 pounds per square inch. Tobin bronze contains 
58.2 per cent, copper, 2.3 per cent, tin, and 39.5 per cent, 
zinc. 

Art. 67. — Cement — Cement Mortar — Concrete. 

The ultimate compressive resistances of mortars and 
concrete determine the carrying power of many engineering 
works, and it is of much importance to ascertain those 
resistances and the conditions under which they may be 
made the greatest possible. Obviously, the carrying power 
in compression of both mortars and concretes will depend 
upon a considerable number of elements such as the character 
of the cement, the proportions of mixture of the sand and 
cement for mortar or of the cement, sand, and gravel or 
broken stone for concrete, the thoroughness of the ad- 
mixture, the amount of water used, the conditions under 
v/hich the mortar and concrete are maintained while the 
operation of setting is taking place, the temperature, and 
other various influences. 

The modulus of elasticity of concrete must necessarily 
depend chiefly upon the proportions of the mixture and the 
age of the concrete when tested. It will also depend to a 
material extent upon the intensity of compressive stress at 
which the strain is observed. At this point a clear under- 
standing of the elastic behavior of the mortars and concrete 
is necessary to a correspondingly clear view of what takes 
place in a concrete-steel beam under loading. In many 
cases of concrete under compression of varying intensities 
a careful measurement of the resulting strains shows that 
a permanent deformation or compression remains at least 
for the time being after the removal of the load, even 
when the latter is sometimes not more than 100 or 200 
pounds per square inch. This permanent set is dependent 
upon the age of the material and usually, perhaps always, 



396 COMPRESSION. [Ch. VIII. 

decreases as age increases. In many other cases a per- 
manent set is observable only under intensities of stress 
as high as looo or 1200 pounds per square inch, or even 
considerably more. When these sets occur they are fre- 
quently found far below what may probably be termed the 
elastic limit of the material, and in some quarters they have 
given the impression that mortar and concrete have little 
or no true elastic behavior. This, however, is an erroneous 
view, as in the testing of concrete and mortar cubes equal 
increments of stress intensities quite uniformly give equal 
increments of strain or deformation over a considerable 
range. Although the upper limit of this essentially constant 
ratio between stress and strain is usually not very clearly 
defined, it is so defined in a considerable percentage of 
cases and in almost all tests of well-made concrete and 
mortar that limit may readily be assigned near enough for 
all practical purposes. 

A large amount of data bearing upon these points will 
be found in the " Report of Tests of Metals and Other Mate- 
rials" at the Watertown Arsenal for 1899. Twelve-inch 
cubes with a great variety of proportions of constituent 
elements ranging from a few days up to six months in age 
were employed in those investigations. Figs, i and 2 ex- 
hibit graphically the results of twelve of those tests so taken 
as to be fairly representative of all. The vertical ordinates 
of the curves represent compressive stress intensities up to 
failure, while the horizontal ordinates represent the total 
compressive strains or deformation imder the corresponding 
stresses also up to the point of failure. These strains are 
shown in the figures one himdred times their actual amoimts. 
In Fig. I the concrete nine days old shows only little resist- 
ing power and a low coefficient of elasticity, as would be 
expected. In nearly all the other cases, on the other hand, 
the ratio between stress and strain is reasonably constant 



Art. 67.] 



CEMENT— CEMENT MORTAR— CONCRETE. 



397 



Up to nearly 1000 poiinds per square inch. The two excep- 
tions are found in Fig. 2, belonging to i to 3 Portland- 
cement mortar and to i, 2, and 4 steel-cement concrete, the 
former four months old and the latter three months old. 





























/ 




































# 












3000 


PDSt 














3000 


PDSt 




fi 
.•>1 


















.*^ 


■<:. 


''t>' 


wos. 


iios.^ 


¥ 
















2000 




N / 


^^ 


U^ 






2000 






^ 


>« 








c 


// 










o^^^ 


. 


// 


^ 
















1000 








s^^^ 


25^ 


3>i 




1000 


■•/ 


V 




















^ 












"/ 






































/ 
























.0 


}5 


.0 


1 


.0 


5 IN 


..! 


3. I 


.0 


35 


.C 


1 


.0 


15 IN 
























5000 


PDSt 






































/ 
















4000 






















/ 


























0^ 


ejiS^ 






^ 1 


/ 
















3000 








P^- f 


^ 






qnoo 


























^;> 


LX 












i 


















3000 




I 


/ 












..// 
























,.^OS 










t/ 




















1000 




I'j: 


.^ 


::-^ 










■7 


<N.^ 


0^ 


^3:^ 
















> 














/ ^" 


^ 




















r 
















1 





















.003 .01 .015 IN. .005 .01 .015 in. 

Fig. 2 



On the other hand, the 1,2, and 4 concrete six months old 
in the right-hand group of Fig. i discloses constant propor- 
tionality between stress and strain up to 2000 pounds per 
square inch, and the same observation may apply to a sim- 



398 COMPRESSION. [Ch. VIII. 

ilar concrete represented by one of the curves in the left- 
hand group of Fig. 2 . Again the i to i granite-dust mortar 
four months old represented by one of the curves in the 
right-hand group of Fig. 2 shows a constant ratio up to 
nearly 4000 pounds per square inch. Indeed, the whole 
group of curves probably show^s a more satisfactory approach 
to a constant ratio between stress and strain than do similar 
curves for cast iron. It should be stated, as will be observed 
by referring to the report cited, that some of the curves 
shown in Fig. i and Fig. 2 belong to groups for w^hich small 
permanent sets were observed below elastic limits, while 
others belong to those which show no such permanent set. 
This observation does not appear from the test records to 
be applicable to any particular character of curves, but 
.sometimes to those which are more nearly straight and some- 
times to those which are less so. 

The results deduced from the tests of cubes covered by 
the 1899 and other "Reports of Tests of Metals and Other 
Materials ' ' are confirmed by the investigations of such for- 
eign authorities as M. Considere, ]\Ielan, Brik, and others. 
They show conclusively that it is reasonable and safe to 
apply to concrete and concrete-steel beams the formulae 
established by the common theory of flexure after intro- 
ducing into_ them empirical quantities established by experi- 
ment precisely as is done with iron and steel beams. 

Table I is a condensed statement of average values of 
the modulus of elasticity for concrete of different propor- 
tions of mixture prepared by Mr. Edwin Thacher from 
original sources, including the annual Reports of Tests of 
Metals and Other Materials carried on by U. S. officers at 
the Watertown Arsenal for a lecture given by him at the 
College of Civil Engineering of Cornell University, 1902. 

This table exhibits as reasonable values for the coeffi- 
cient of elasticity in compression as can be determined at 



Art. 67. 



CEMENT— CEMENT MORTAR— CONCRETE. 



399 



f fe 



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w. 
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GO xn 

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rt o 



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o o c o 



r^ ro r-- O 

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088 

_ q_ q_ c 

ro C" t~^ ^ r^ 
CO C O lOvO 

OC SC O 1- CO 



000 

88 ■ 



O C O G O 
O C O O O 

o c o o o 



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cox oc vC ^ 



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89050 

o, o. o, q^ o^ 

r^ r^ rf cT w" 
O vc vO i^ cs 
vO vo CO Tl- to 



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r^ CO 10 10 10 

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vO ►- 00 X 1-1 



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t^ O X ^ o 
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hT oT cT >-<" i-T 



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mx X u^vo 
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04" i-h" hT hT i-T 



0000 

8888 



« H^ O H-. CO 

MD C^ O vO O 

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r^ '^ CO CO CO 



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r--vo 04 r-^ o 
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ro ^ CO CO CO 



CO w 04 w w 
r^ r^ to cox 
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o__ o o o o 

-^ -^ O" w rC 
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04 04 t^ O to 

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04 0) VC vo T:f 

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d 






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1 : : : : 






















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• • c 




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II 


5 
1 

^ 


tr 
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Alpha. 
Germa 




Atlas. 
Alpha. 
Germa 
Albert 




Atlas. 

Alpha. 

Germa 

Alsen. 

Mean. 



400 COMPRESSION. [Ch. VIII. 

the present time. The value to be selected for any particu- 
lar case will depend upon the proportions of mixture and 
upon the degree of balancing of the sand and gravel or 
broken stone, although the influence of the latter cannot 
be definitely stated. It is not improbable that a -considera- 
ble portion at least of the variations in the results of the 
table are due to the varying degrees of natural balancing 
in the different test blocks. The value will also depend 
upon the age of the concrete. For all ordinary engineering 
constructions it is reasonable to take the coefficient of com- 
pressive elasticity at 2,500,000 to 3,000,000 pounds per 
square inch for a concrete mixture of i cement, 2 sand, and 
4 gravel or broken stone. This table shows that practi- 
cally the same value may be taken for a concrete of i cement, 
3 sand, and 6 gravel or broken stone, especially if the mate- 
rials are well selected and balanced. If the concrete is 
mixed in the proportions of i cement, 6 sand, and 1 2 gravel 
or broken stone, the coefficient of elasticity is seen to 
decrease materially and should not be taken higher than 
1,500,000 pounds per square inch. Suitable quantities 
for mixtures other than those named in the table can be 
reasonably and safely selected from those afforded in it. 

These values show that the ratio of the coefficient of 
elasticity for steel over that for concrete may range from 
10 to 20 for the varying conditions described. 

The more common practice is to make this ratio 15, i.e., 
on the basis of 30,000,000, for the modulus of elasticity for 
steel and 2,000,000 for concrete. The ratio of 12, however, 
is sometimes found by taking the same value as before for 
the modulus of steel, but 2,500,000 for the modulus of 
elasticity for concrete. The ratio of the two moduli is 
constantly used in the treatment of reinforced concrete 
work. 

A further consideration must be kept in view in con- 



Art. 67.] 



CEMENT— CEMENT MORTAR— CONCRETE. 



401 



nection with the vakie of the modulus of elasticity for con- 
crete, and that is the fact alluded to in previous pages that 
nearly all concrete and reinforced concrete work must 
usually, carry considerable loading, in the exigencies of con- 
struction, when it has attained no greater age than perhaps 
10 to 30 days, i.e., before the m.odulus of elasticity (or ultimate 
resistance) has attained its full value. Again, a large mass 
of concrete, as actually built, cannot reasonably be expected 
to have as high a modulus as 12 -inch cubes or other com- 
paratively small pieces made and tested in a laboratory. 
For all these reasons it is prudent to take a rather low value 
of the modulus of elasticity for the analytic work of design. 
The following tabulated statement shows ultimate resist- 
ances per square inch of 12 -inch cubes of concrete obtained 
in the Testing Laboratory of the Department of Civil 
Engineering of Columbia University in 191 2 by Mr. James 
S. Macgregor, in charge of the laboratory. 

GRAVEL CONCRETE; i Cement, 2^ Sand, 5 Gravel. 



Ult; Resistance Pounds per Sq. In. 



Max. 



Mean. 



Min. 



Alsen . . . . 
Atlas. . . . 
Atlas . . . . 
Iron Clad 
Iron Clad 
Lehigh. . . 
Lehigh . . . 
Vulcanite 
Vulcanite 
Alsen . . . . 
Alsen*. . . 



1,917 
1,905 
2,223 

1,789^ 

1,848 

2,717 

2,278 

1,162^ 

1,735 
2,322 
1,202 



1 = 773 

1,796 

2,191 

1,553* 

1,778 

2,584 

2,139 

1,097* 

1,593 
2,088 
1,023 



1,557 

1,706 

2,152 

1,431* 

1,657 

2,431 

2,007 

1,047* 
1,518 
2,006 
944 



Age of 

all 
cubes 

42 
days 



* Gravel unwashed. 



The coarse aggregate for all cubes was river gravel with 
stones up to i-inch size. Some of the gravel contained an 
excessive amount of dirt or other fine material, which 



402 



COMPRESSION. 



[Ch. VIII. 



Table II. 

MEAN ULTIMATE COMPRESSIVE REvSISTANCES OF 12-INCH PORT- 
LAND-CEMENT CONCRETE CUBES. 







Mean Ultimate Resistance, 


Coefficient of Elasticity in 


Portland Cements. 


Pounds per Square Inch 
at Atre. 


Pounds per Square Inch 
at Age. 


Brand; C 


jmposition. 










7 Days. 
1,724 


I Mo. 


3 Mos. 


6 Mos. 


I Mo. 


3 Mos. 


6 Mos. 


\ I 


c, 2 s., 4 b. St. 


2,238 


2,702 


3,506 


2,500,000 


3,571,000 


5,000,000 


Say lots. . ■{ I 


c, 3 s., 6 b. St. 


1,625 


2,568 


2,882 


3,567 


2,778,000 


4,167,000 


2,500,000 


1 I 


c, 6 s.,i2 b.st. 


67s 


800 


1,128 


1,542 


833.000 


2,273,000 


2,083,000 


\ 1 


c, 2 s., 4 b. St. 


1,387 


2,428 


2,966 


3,953 


3,125,000 


4,167,000 


3,125,000 


Atlas . . . ■< I 


c, 3 s., 6 b. St. 


1,050 


1,816 


2,538 


3,170 


3,125,000 


2,778,000 


3,571,000 


i 1 
f I 


C, t) S.,I2b.St. 
c, O S., 2 b. St. 


.■594 


1,090 


1,201 


1,583 


1,316,000 


1,136,000 


1,786,000 


3,294 














Alpha. . . \ I 


c, 2 s., 4 b. St. 


902 


2,420 


3,123 


4,411 


2,083,000 


4,167,000 


3,125,000 


c, 3 s., 6 b. St. 


892 


5,150 


2,355 


2,750 


2,083,000 


3,571,000 


4,167.000 


c, 6 s.,i2 b.st. 


564 


1,218 


1,257 


1,532 


1,667,000 


1,786,000 


1,923,000 


f; 


c, O S., 2 b. St. 

c, 2 s., 4 b. St. 


2,734 


3,246 
2,642 


3.858 
3,082 


5,129 
3,643 


3,571,000 


2,778,000 


3,571,000 
4,167,000 


Germania -i 








c, 3 s., 6 b. St. 


1,550 


2,174 


2,486 


2,930 


2,273,000 


2,778,000 


3,125,000 




c, 6 s.,i2 b.st. 


759 


987 


063 


815 


961,000 


2,083,000 


1,786,000 




c, O S., 2 b. St. 


3,118 


3,240 


3,710 


5,332 


2,273,000 


2,273,000 


3,571,000 


Alsen ...-il 


c, 2 s., 4 b. St. 


1,592 


2,269 


2,608 


3,612 


2,788,000 


2,778,000 


4,167,000 


c, 3 s., 6 b. St. 


1,438 


2,114 


2,349 


3,026 


2,273,000 


2,778,000 


3,571,000 




c, 6s.,i2 b. St. 


417 


873 


844 


1,323 


1,562,000 


1,562,000 


1,786,000 



10-INCH CUBES. 



Alpha. . . I c, o s., 2 b. st 







5,463 


6,556 







5,000,000 



In this table each ultimate resistance is a mean of four to six tests. 



Table III. 

MEAN ULTIMATE COMPRESSIVE RESISTANCES OF 12-INCH PORT- 
LAND-CEMENT CONCRETE CUBES WITH LOAD TAKEN ON 
8'^ BY 8''.25 PLATE ON ONE FACE. 



Portland Cements. 
Brand; Composition. 


Mean Ultimate Resistance, 

Pounds per Square Inch 

at Age. 






I Month. 


3 Months. 


6 Months. 




• 11 ^ I c, S., 2 b. St... 
Alpha--- ]." 2" 4 " ... 
Germania -;;?,- °?,-; ^^- 

A.sen....|;f,-.°?;.4\^t;;; 


5,089 
3.287 
4,327 
3,587 
4,087 
3.233 


4.531 
3.522 
3,426 


5,669 
6,671 

4,582 
6,382 
4,983 


Each ultimate 
resistance is a 
mean of three 
tests. 




A view exhibiting the failure under compression of a i2-in. concrete cube. The 
composition is I Portland cement, I sand, and 4.5 broken stone. The age of 
the concrete was I year, 8 months, 23 days, and the ultimate compressive 
resistance attained was 4481 lbs. per sq. in. 

{To face page 402.) 



Art. 67.] CEMENT— CEMENT MORTAR—CONCRETE. 403 

accounts for the low values of the starred ultimate resist- 
ances per square inch, as indicated by the footnote. The 
age of all the cubes was 42 days, also as indicated in the 
table. These results are unusually valuable in one respect, 
in that the cubes were not mixed in the laboratory, but in 
the field, where actual work was being done, and hence 
received no special care in the operation. 

Tables II and III contain the results taken from the 
" U. S. Report of Tests of Metals and Other Materials " for 
1899. They exhibit the ultimate compressive resistances 
of cubes of Portland-cement concrete, the cements being 
among the well-known brands. The ages of these cubes 
vary from seven days to six months. The data show 
clearly the increase of ultimate resistance with the ages of 
the cubes, and the same observation applies to the three 
columns showing the coefficients of elasticity at one month, 
three months, and six months. The compositions of the 
different concretes of Table II are those quite generally 
employed in engineering practice. 

Table III exhibits the ultimate resistances of the same 
concretes, but with the pressure applied to the 12-inch 
cubes on areas 8 inches by 8^ inches, this end being at- 
tained by the use of steel plates. As would be expected, 
the ultimate resistances are seen to be considerably greater 
than are found with the total load distributed over the 
entire surface of a cube. 

The broken stone used in the cubes, the results of whose 
tests are given in Tables II and III, was a conglomerate from 
Roxbury, Mass., and the sand was coarse, clean, and sharp. 
The voids of the broken stone measured 49.5 per cent, of 
their total volume. 

Table IV, taken from the same volume of the '' U. S. 
Report of Tests of Metal and Other Materials'' as Tables II 
and III, exhibits the ultimate compressive resistances of 



404 



COMPRESSION. 



[Ch. VIII. 



Table IV. 

MEAN ULTIMATE COMPRESSIVE RESISTANCES OF MORTAR AND 
CONCRETE 12-INCH CUBES. 



Brand; Composition. 



Mean Ultimate 

Resistance, 

Pounds per 

Square Inch 

at Age of 

Four Months. 



Weight per 

Cubic Foot, 

Pounds. 



Coefficient of 

Elasticity, 

Pounds per 

Square Inch. 



Alpha 
Portland 



f I c, I s., o b 

" 2 " O 



^I 



Li 



Atlas 

Portland 

Star 

Portland 

Saylors 

Portland 

Germania I 

Portland 

Alpha 

Portland 

Steel slag 






r 

I 
I 



Hoffman 
Rosendale 

Norton \ ^ 

Rosendale ) 



St. 



s., o 
" 4 



o 

2 " O 

2 " 4 



4,371 

2,506 

1,812 

829 

484 

185 



5,570 

5,045 

3,979 

4,353 

5,306 

1,743 
1,939 t 

741 

643 
277 t 
332 t 



136.5 
134-2 
133-8 
120.9 

119-5 
116. 9 
III. 5 

141.5 
134-5 
134-7 
134-7 
137-3 

126.6 

152. 1 t 

X27.7 

125.2 
120.7 

146.2 t 



3,571,000 
3,125,000 
1,786,000 



6,250,000 

4,167,000 

3,125,000 

2,500,000 

3,571,000 

1,190,000 
2,500,000 



* Granite dust. 



t Age, 3 months. 



X Trap rock, broken stone. 



Table V. 
CHEMICAL ANALYSES OF PORTLAND AND STEEL-SLAG CEMENTS. 



Cement. 


Silica. 


Oxide 
of Iron. 


Alumina. 


Lime. 


Magnesia. 


Sulphur 
Trioxide. 


Carbon 
Dioxide. 


Alpha. . . 


20 


2.8 


10.87 


58.66 


3-35 


1-34 


2.56 


Star. . . . 


21.73 


2-5 


9-47 


56.34 


3-61 


1. 91 


3-94 


Standard 


22.5 


2.6 


1 1 . 98 


51-44 


3.61 


1-57 


5-96 


Alsen. . . 


20.67 


2 . I 


14.6 


42. 16 


2.32 


2.32 


4-45 


Steel. . . . 


31 .02 


Trace 


10.9 


57-31 


4-05 


3.36 


4.81 



Art. 67. 



CEMENT— CEMENT MORTAR— CONCRETE. 



405 



the mortar and concrete 12-inch cubes described therein. 
These results need no explanation, as they are similar to 
those which have already been given, but it is well to note 
that the last four lines of the table give results belonging to 
two brands of natural cement. There are also shown one 
test of a steel-slag cement mortar cube and one of concrete. 
Table V exhibits the chemical analyses of the Portland 
and steel-slag cements named in Table IV. These analyses 
exhibit about the usual composition of the various grades 
of cement to which they belong. 

Table VI. 
COMPRESSION TESTS OF 12-INCH CUBES OF PORTLAND-CEMENT 













CINDER CONCRETE. 








Brand. 


Composition. 


Age 

when 

Tested, 

Days. 






Ultimate Resistance 
in Lbs. per Sq. In. 


Coefficient 

of 

Elasticity, 

Pounds. 




Max. 


Mean. 


Least. 


Germania . 


I c. 
I c. 
I c. 
I c. 
I c. 
I c. 
I c. 
I c. 
I c. 


, I s 
, 2 S 
, 2 S 
, 2 S 
, 3S 
, I S 
, 2 S 
. I s 
, 2 S 


, 3 cir 
, 3 
, 4 
, 5 
, 6 
, 3 
, 5 
, 3 
, 5 


der 


99 and 102 
102 

98 
98 and loi 

91 

90 

90 

90 

90 


3 
3 
3 
3 

I 

3 
3 

3 


no. 4 

112. 8 

107.9 
106.3 

103.5 
114. 1 

no 
116. 3 
109.9 


2,023 
1,701 
1.344 
1,114 

Hi 
2,988 

1,715 

2,580 

1,263 


2,001 
1,634 
1.325 
1,084 
788 
2,834 
1,600 
2,414 
1,223 


1,975 
1,589 
1.295 
1,052 

2,780 
1,402 

2,295 

1,200 





Alpha....*. 
Atlas.*.*.'.' : 


2,500,000 

1,279,000 

3,125,000 

857,000 



The results exhibited in Table VI are interesting as 
belonging to Portland-cement cinder concrete and they are 
of- practical importance because such concrete is used in 
many buildings especially for floors, in consequence of its 
weighing much less than ordinary broken-stone concrete. 
The ages of these cinder concrete cubes is seen to run from 
90 to 102 days, which is sufficient to give nearly the full 
ultimate resistance of such material. It is seen, however, 
that cinder concrete is materially less strong or capable of 
ultimate compressive resistance than either broken-stone 
or gravel concrete having the same proportions of mixture 
in its composition. The column giving the weight in 



4o6 



COMPRESSION. 



[Ch. VIII. 



pounds per cubic foot shows that cinder concrete weighs 
but about three fourths as much as that made with gravel 
and broken stone. The data contained in this table were 
taken from the '' U. S. Report of Tests of Metal and Other 
Materials" for 1898. 

Messrs. Harold Perrine, C.E. and George E. Strv^nan, 
C.E. presented a paper to the Am. Soc. C. E. in 191 5 
describing their extended investigation* in " Cinder Con- 
crete for Floor Construction between Steel Beams." The 
Table VII is taken from that paper and each value is a 
mean of ten results, except those in the second column 



Table VIL 





C. S. Cin. 
1:2:5 
Continuous 

rnixer. 
Coltrin. 

Alsen. 


C. S. Cin. 
1:1:5 
By hand 

turned 

twice. 
Dragon. 


C. S. Cin. 
1:2:5 

Batch 

mixer. 

Vulcanite. 


C. S. Cin. 


Method 

Cement . 


Ransome. 
Mixer, 

Atlas. 






Sand 


Long Island Bank Sand, North Shore. 




Anthracite. 


Cindeis. 


Ice 
plant. 


Local 
hotel 
steam 
plant. 


Local. 


Local office 

building 

steam 

plant. 


Weight, lbs. per cu. ft 

One month test: 

Ult. Resist., lbs. per sq. in. 

E, lbs. per sq. in 

Two months test: 

Ult. Resist., lbs. per sq. in. 

E, lbs. per sq. in 

Six months test: 

Ult. Resist., lbs. per sq. in. 

E, lbs. per sq. in 


107 

407 
924,600 

701 
1,134,000 

933 
971,000 

913 
993,000 


100 

507 

857,400 

662 
1,030,000 

754 
1,050,000 

813 
956,000 


107 

818 
1,230,000 

1,254 
1,740,000 

1,744 
1,348,000 

1.465 
1,200,000 


109 

980 
1,492,000 

1,035 
1,428,250 

1,478' 
1,276,000 

1,475 
1,320,000 


One year test: 

Ult. Resist., lbs. per sq. in. 
E lbs per sq in . . 







* Made in the testing laboratory of the Dept. of Civil Engineering, Col- 
umbia University by the aid of the Wm. R. Peters, Jr. memorial research fund. 



Art. 67.] CEMENT— CEMENT MORTAR— CONCRETE. 407 

from the right side of the Table, which are means of nearly 
that number. The compressive test specimens v/ere cinder- 
concrete cylinders 8 inches in diameter and 16 inches long. 
The values given in the Table are representative of good 
structural cinder concrete. 

A large number of tests, the results of which need not 
be given here, have shown that gravel may advantageously 
be used, in the interests of economy, in the place of broken 
stone for concrete. On the whole, the broken-stone concrete 
is probably stronger than that made with gravel, but the 
difference is not material for all ordinary cases. The 
gravel should not be water-worn, but have sharp, gritty 
surfaces to which the setting cement may strongly bond 
itself. All sizes from the largest permissible down to 
coarse sand should be taken, and when so balanced the 
voids may be reduced as low as 20 per cent, of the total 
volume of the gravel or even lower. This balancing 
of the broken stone or gravel enhances both economy 
and resisting qualities. 

A careful examination of all the Tables, I to V, shows 
that reasonably well-made broken-stone concrete may 
carry a load of 300 to 500 pounds per square inch without 
exceeding i to |, or possibly |, of its ultimate resistance, 
the composition of the mixture being i cement, 2 sand, 
and 4 broken stone, or perhaps i cement, 3 sand, and 5 
broken stone. It is possible that this may be an under 
statement of the capacity of the concrete if the mixture is as 
well balanced as it should be. It is a mistake, as has been 
shown repeatedly by actual test, to screen out the finer 
portions of the broken stone or to attempt to secure an 
approximately even sand grain. It is conducive to an 
increased resistance as it is to increased economy to balance 
the sand, gravel, or broken stone by using all the varying 
sizes between the least and the greatest. Indeed, in many 



4o8 



COMPRESSION. 



[Ch. VIII. 



Table VIII. 
COMPRESSIVE RESISTANCES OF 12"xl2' 



CONCRETE COLUMNS. 



.c*; 


Age. 
Days. 




W'ght 


Ult. 






Composition. 


in Lbs. 
per 


Resist, ir 
Lbs. per 








Cu.Ft. 


Sq. In. 








I cement, 3 sand, 4-1^" | 




. 




2 


47 ) 


broken stone, 2-Y' broken > 


145 


1,072 




2 


47 1 


stone ) 


145 


917 




4 


47 


do. 


144 


1,067 




4 


47 


do. 


144 


1,132 




6 


46 


do. 




844 




6 


46 


do. 


143 


1,048 




8 


42 


do. 


145 


935 


Hand mixed 


8 


42 


do. 


145 


900 




lO 


40 


do. 


142 


909 




lO 


41 


do. 


143 


807 




12 


39 


do. 


144 


947 




12 


39 


do. 


144 


980 




14 


34 


do. 


145 


936 




14 


35 


do. 


145 


907 J 






A-( 


I cement, 3 gravel, 4-1 i'' i 




1,185 




2 


VA 


broken stone, 2-|" broken ,'- 


145 




2 


4/| 


stone ) 


147 


1,183 




4 


48 


do. 


143 


980 




4 


48 


do. 


144 


936 




6 


48 


do. 


146 


1,131 




6 


48 


do. 


146 


1,200 




8 


42 


do. 


146 


1,108 




8 


42 


do. 


146 


1,086 




lO 


41 


do. 


146 


1,015 




lO 


42 


do. 


146 


1,000 




12 


37 


do. 


149 


1,400 




12 


39 


do. 


148 


1.500 




14 


35 


do. 


148 


858 




14 
6 
6 


35 

42 ( 

42 1 


do. 
I cement, 6 gravel, 8-1^" 
broken stone, 4-I" broken ■ 
stone 


148 

143 
144 


807 

500 
467 


Machine mixed 


6 
6 




I cement, 7 gravel, 8f-ii" 








42 

42' 


br'k'n stone, 4^-f'' br'k'n - 
stone ) 


141 
142 


427 
436 




6 
6 




I cement, 5 gravel, 6f-ii" 


146 


708 




45 
45" 


br'k'n stone, 3^-!" br'k'n Y 
stone 


146 


747 




6 
6 


46 ' 
46 


I cement, 4 gravel, 5^-1^" ) 


146 


900 




br'k'n stone, 2§-f" br'k'n V 
stone ) 


145 


797 




12 


36 ■ 

39 


I cement, 3 gravel, 6— |" [ 


150 


1,250 


"] Reinforced with 


12 


broken stone ) 


149 


1,700 , 


(^ 4-f" cold-twisted 












j steel rods embed- 












J ded in the concrete 




( 


I Silica Portland cement, 2 ) 








9 


580^ 


coarse clean sand, 3 quartz >• 


148 


2,548 






( 


gravel (^'-2") ) 









Art. 68.] BRICKS AND BRICK PIERS. 409 

cases it may be advisable to use the entire product of the 
crusher. 

The relation between the ultimate compressive resist- 
ance of concrete made with balanced material and the 
lenofth of column is illustrated by the results given in Table 
VIII, which has been collated and arranged from the " U. vS. 
Report of Tests of Metal and Other Materials "for 1897. 
The heights of column range from 2 to 14 feet. While there 
are some exceptions, the rule is general that, other things 
being equal, the ultimate resistance decreases as the length 
or height of column increases. On the whole, the machine- 
mixed material appears to be a little stronger than the 
hand-mixed, but the difference is not substantial except for 
the 8, 10, and 12 feet lengths. 

Art. 68. — Bricks and Brick Piers. 

The ultimate compressive resistance of bricks depends 
largely upon the manner in which they are tested and the 
care with which the surfaces pressed are filled out with a 
proper cushion and made truly parallel to the bearing 
surfaces of the testing machine. The best of bricks as 
produced for the market do not have opposite faces truly 
parallel, and hence when they are placed in a testing 
machine for testing to failure the pressure will be con- 
centrated at different points and the bricks will be broken 
partly by bending before the full ultimate compressive 
resistance is developed unless the pressed surfaces are 
made true by some kind of a ciishion. This cushioning is 
frequently and perhaps usually done with plaster of pans, 
as in the case of the tests of bricks at the U. S. Arsenal, 
Watertown, IMass., the results of which are given in Table 11. 
Again, a brick tested on edge will give a less ultimate 
resistance per square inch than when tested fiat and the 



4IO COMPRESSION. [Ch. VIII. 

resistance on end per square inch of section will be less 
than that on edge. When the brick is tested flatwise, 
even when truly surfaced with a cushion such as plaster of 
paris, it is a very short block and the friction of the pressed 
surfaces on the bearing faces of the testing machine is 
sufficient to give the compressed material substantial lat- 
eral support, not permitting it to separate and crush away 
readily. It will be found, therefore, that when blocks are 
tested flatwise the ultimate resistances per square inch, 
as a whole, will be much higher than when tested on edge. 
This condition of things holds to some extent when the 
bricks are tested on edge, so that an endwise test will give 
the ultimate compressive resistance per square inch some- 
what less than that found when the brick is tested on edge. 
An endwise test of the brick more truly represents the 
ultimate compressive resistance of the material than a test 
either flatwise or on edge. 

A series of tests of a variety of bricks and terra-cotta 
made in 1896 at the U. S. Arsenal at Watertown, Mass., 
gave moduli of elasticity about as follow^s: Pressed brick, 
1,000,000 to 3,000,000 pounds per square inch, the hardest 
varieties giving the higher values and the softer material, 
the lower values; hard buff brick and terra-cotta, 4,000,000 
to 4,800,000 pounds per square inch. Some soft-face brick 
gave moduli of elasticity varying from about 400,000 to 
890,000 pounds per square inch. These determinations of 
the modulus were made with intensities of pressure from 
about 1000 to 4000 or 5000 pounds " per square inch. 
Such experimental results ordinarily show some erratic or 
abnormal features and these tests were no exception to 
that rule. 

The coefficients of thermal expansion and contraction 
per degree Fahr., were at the same time found to range 
from .00000205 to .00000754, the larger of these values 




A solid i6-incli square-face brick pier laid in lime 
mortar It was tested at the U. S. Arsenal, Water- 
town, Mass., and gave an ultimate compressive 
resistance of 1337 lbs. per sq. in. The pier is 
shown as it existed after failure. 

{To face page 410.) 



Art. 68.1 



BRICKS AND BRICK PIERS. 



411 



being about 25 per cent, higher than the coefficient for 
concrete. 

In the Proceedings of the Am. Soc. C. E. for March, 
1903, Mr. S. M. Turrill, Assoc. Am. Soc. C. E., gives the 
results of a large number of tests of common building 
brick, 2 in. by 4 in. by 8 in. in size, manufactured at Horse- 
heads, N. Y. The following table is fairly representative 
of the results of Mr. Turrill's tests, made with great care 
at the civil -engineering laboratories of Cornell University: 



TEST OF COMMON BUILDING BRICK. 



Brick Tested. 


No. of Tests. 


Ultimate Compressive Resistance, 
Pounds per Square Inch. 




Greatest. 


Mean. 


Least. 


On end 


12 
12 
12 


3.763 
3,913 
5,463 


2,628 
2,832 
3,995 


1,234 
1,897 
2,665 


On edge 


Flat 





These bricks were tested in their natural condition as 
delivered from the kiln ready for use. 

Other tests were made of the same brick saturated with 
water and after being reheated in a suitable oven. This 
latter test was designed to disclose the quality of brick 
after having passed through a conflagration. The satu- 
rated bricks tested on end and on edge showed material 
loss of resistance below that of their natural condition, but 
those tested flat showed large gains. The reheated bricks 
exhibited large gains in all three modes of testing. These 
bricks were obviously not of hard-burned, high-resisting 
character. 

The coefficient of elasticity of twelve of these bricks 
ran from 540,000 to 1,815,000 pounds per square inch, with 
a mean value of 1,305,000 pounds. 



412 



COMPRESSION. 



[Ch. VIII. 



A large number of determinations of the ultimate com- 
pressive resistances of bricks were made among the earlier 
experimental investigations at the U. S. Arsenal at Water- 
town, Mass. These results showed values for hard-burned 
bricks varying from about 8,000 to about 12,000 pounds 
per square inch with an average of about 9,000 pounds per 
square inch when tested on edge. What may be termed 
medium bricks, i.e., intermediate between hard-burned 
strongest bricks and common building bricks, gave results 
varying from about 4,000 to about 8,000 pounds per square 
inch, with an average value of about 5,500 pounds per square 
inch w^hen tested on edge. 

The following results of tests of three different kind 
of brick and hollow tile were obtained by Mr. J. S. Mac- 
gregor in the testing laboratory of the Department of Civil 
Engineering at Columbia University. The ultimate resis- 
tances given are the means of seven sets of tests, eight in 
each set. Half bricks were tested flatwise. This mode of 
testing obviously yields much higher values than if the 
bricks were tested on edge. 



Lbs. per sq. in. 



Max. 



Mean. 



Min. 



Common Hudson River, moulded 

Stiff Clay, side cut 

Harvard, over-bumed . 



4.357 
2,537 



3,203 
2,305 
6,642 



2,006 
2,072 



The hollow tiles were of two types, six-core and two-core. 
The cross-sections were 10 inches by 12 inches, 8 inches by 
12 inches, 8 inches by 16 inches, and 12 inches by 12 inches. 
The length or height of each set of tiles was 12 inches with 
one exception of 8 inches. The tiles were all tested with 
the webs (or cores) vertical and the net sectional areas 



Art. 68. 



BRICKS AND BRICK PIERS. 



413 



varied from about 41 square inches to 60 square inches. 
The ultimate resistances per square inch on both the net 
sections and the gross sections are as given below. There 
were five sets of ten tests each and the results given are 
the greatest, mean and least results of the five sets. 





Lbs. per sq. in. 




Max. 


Mean. 


Min. 


\et section 


5.718 
2,680 


4.598 
2,090 


3,826 
1,710 


Gross section 





Brick Piers. 

Inasmuch as tests of brick piers have shown that 
their ultimate compressive resistances run only from about 
1000 to 4500 pounds per square inch, depending upon 
the character of the mortar, it is seen that in such masonry 
a small portion only of the compressive resistance of the 
bricks is developed in piers and other similar brick-masonry 
masses. 

These latter results doubtless depend largely upon the 
cementing material. There is no question that the ulti- 
mate resisting capacity of brick masonry is affected 
greatly by the resisting capacity of the mortar, and 
the same general observation can be applied to other 
classes of masonry. There is more than this, however, 
affecting the carrying capacity of brick and other grades 
of masonry as compared with the ultimate compressive 
resistance of the bricks used in the one case of masonry 
or of the individual stones employed in the other. The 
texture or character of the mass of burned clay com- 
posing the brick is exceedingly variable, both in conse- 
quence of the varying mixture of the material in the bricks 



414 COMPRESSION. [Ch. VIII. 

before being burned and in consequence of the varying 
degree of burning in each individual brick. Again, what- 
ever may be the care in placing the bricks in a testing- 
machine, including the cushioning of the ends, it is prac- 
ticably impossible to secure anything like a uniform bear- 
ing upon either the ends, sides, or beds. Their irregular 
dimensions and exterior surfaces and the varying quality 
of the materials, even in the best of brick, introduce into 
their resisting capacity elements of variation which are 
frequently so great as to lead to abnormal results. While 
the mortar used in forming a mass of brick masonry im- 
doubtedly fills up many irregularities of surface, voids 
of considerable magnitude frequently remain unfilled. 
The consequence of these uncontrollable elements in a 
mass of brick masonry is always a material reduction of 
ultimate carrying capacity and frequently a large reduc- 
tion. However excellent in quality, therefore, the mor- 
tar or binding materia' in a brick-masonry pier may be, 
it is inevitable that there will be not only a wide range in 
ultimate compressive resistance, but in all cases a material 
reduction below that exhibited by the individual bricks 
when tested by themselves. 

Profs. Arthur N. Talbot and Duff A. Abrams reported, 
in Bulletin Xo. 27 (1908) of the University of Illinois, the 
results of a series of sixteen tests of brick piers and the 
same number of hollow terra-cotta block piers. Two grades 
of brick were used, a hard-burned shale brick and a soft 
under-burned clay brick. Eighteen of the former tested 
on beds gave : 





Lbs. per sq. in. 




Max. 


. Mean. 


Min. 


Ult. Comp. Resist 


14.150 


10,690 


7.030 



Art. 68. J BRICKS AND BRICK PIERS. 

Sixteen of the soft bricks similariy tested gave: 



415 





Lbs. per sq. in. 




Max. 


Mean. 


Min. 


Ult. Comp. Resist 


5.670 


3.920 


2,190 



The hollow terra-cotta blocks were about 4 inches by 
8 inches, 4 inches by Sj inches and 4 inches by 8j inches 
in cross-section, the height or length being generally 8 inches, 
but 4 inches in some cases. These blocks had three cores, 
two i| inches square each and one i^ inches by J inch. 



Table I 
AVERAGE VALUES FOR BRICK COLUMNS 



Columns. 





Ratio of 


Ratio of 


Average 


Ultimate 


Ultimate 


Ultimate. 


of Col- 


of Col- 


Load, lb. 


umn to 


umn to 


per sq. m. 


Ultimate 


Ultimate 




of Brick. 


of "A" 



E 

Initial 

Modulus 

of 

Elasticity. 



Num- 
ber of 
Tests. 



Shale Building Brick. 



A-Well laid, 1-3 portland 
cement mortar, 67 days 

Well laid, 1-3 portland ce- 
ment mortar, 6 months. 

Well laid, 1-3 portland ce- 
ment mortar, eccen- 
trically loaded, 68 days. 

Poorly laid, 1-3 portland 
cement mortar, 67 days 

Well laid, 1-5 portland ce- 
ment mortar, 65 days. . 

Well laid, 1-3 natural ce- 
ment mortar, 67 days. . 

Well laid, 1-2 lime mortar, 
66 days 



3365 


•31 


1. 00 


3950 


•37 


1. 18 


2800 


.26 


•83 


2920 


.27 


.87 


2225 


.21 


.66 


1750 


.16 


•52 


1450 


•14 


•43 



4,780,000 

5,025,000 

4,400,000 

3,525,000 

3,250,000 

800,000 

104,000 



Under-burned Clay Brick. 



Well laid, 1-3 portland ce- 
ment mortar, 63 days . . 



1060 



27 



31 



433.000 



4i6 



COMPRESSION. 



(Ch. VIII. 



The brick columns were about 12 J inches by 12^ inches 
in section and 10 feet long. The mortar joints were about 
f inches thick. Failure of these columns took place chiefly 
by vertical cracks through joints and bricks. Table I gives 
the mean results of these tests. 

The characteristics and dimensions of the terra-cotta 
columns or piers and the average results of tests per square 
inch of gross area are given in Table la. 

Table Ia. 

AVERAGE VALUES FOR TERRA COTTA COLUMNS 



Characteristics 
of columns. 



Number 

of 

Columns 

in Average 



Average 

Ultimate 

Unit 

Load 

lb. per 

sq. in. 



Ratio 
Ultimate 

of 
Column 

over 
Ultimate 
of Block 

(Gross 

area). 



Initial 
Modulus 

of 
Elasticity. 



8^X8| in. 
8|Xi3in. 
13X13 in. 



1-2 Portland cement mortar. All well laid and centrally loaded. 



2 


2885 


■83 


2 


3070 


.89 


2 


2955 


•85 



2,194,000 
2,194,000 
2,194,000 



12^ X 125 m. 



1-3 Portland cement mortar, well laid unless noted. 



Central load 



Eccentric load 

Poorly laid, central load . . . 
Poorly laid, eccentric load . . 
Inferior blocks, central load. 
1-5 mortar, central load . . . 



2 


3790 


74 




4300* 


83* 


I 


3470 


(55 


I 


3305 


64 


I 


3IIO 


60 


I 


3050 


59 


2 


3350 


65 



2,765,000 

2,330,000 
3,200,000 
2,500,000 
2,300,000 
2,690,000 



* Estimated. 



The average age of columns when tested was 67 days. 

The joints of the columns were about | inch thick and 
the blocks were laid on end. Failures were sudden and 
accompanied or caused, by longitudinal cracks. In fact, 




An 8 X i6-in.-face brick pier witli i6-iii. square base 
laid in lime mortar. It was tested at the U. S. 
Arsenal, Watertown, Mass., and gave an ultimate 
compressive resistance of 1233 ^^s. per sq. in. on 
tlie upper section and 601 lbs. per sq. in. on the 
lower section. The cracks due to failure are clearly- 
seen. 

i^To face pc.ge 417.) 



Art. 68.1 



BRICKS AND BRICK PIERS. 



417 



the chief manner of fracture of both brick and terra-cotta 
cokimns or piers is by longitudinal cracking. 

Table II exhibits the results of testing piers of brick 
masonry in the Gov^^sting machine at Watertown, Mass. 
It is taken from '^ Ex. Doc. No. 35, 49th Congress, ist 
Session." The dimensions of piers are shown in the table; 
also the kinds of mortar used and the grades of brick. 
The " common " and ** face " brick, both hard burnt, 
were from North Cambridge, Mass. The other bricks 

Table IL 
CRUSHING STRENGTH OF BRICK PIERS. 





Height 


Section 




Weight 


Ultimate 


No. 


of P 


ier, 


of Pier, 


Composition of Mortar. 


per 


Resistance, 




Ft. 


Ins. 


Ins. 




Cu. Ft., 
Lbs. 


Lbs. per 
Sq. In. 


I 


I 


4 


8X8 


I lime, 3 sand. 


137-4 


2,520^ 




2 


6 


8 


8X8 


I " 3 " 


133-5 


1,877 




, 3 


I 


4 


8X8 


I Portland cement, 3 sand. 


136.3 


3,776 


S 


4 


6 


8 


8X8 


I " "3 " 


133-5 


2,249 


^ 


5 
6 


2 
2 






12X12 
12X12 


I hme, 3 sand. 
I " 3 " 




1,940 
1,900 


■| 


7 


10 





12X12 


I " 3 " 


131-7 


1,511 


0) 


8 


10 





12X12 


I " 3 " 


125.0 


1,807 




9 


2 





12X 12 


I Portland cement, 2 sand. 




3,670 


10 


10 





12X 12 


I " "2 " 


132.2 


2,253J 




II 


I 


4 


8X8 


I lime, 3 sand. 


135-6 


2,440^ • 


12 


6 


8 


8X8 


.1 ;| 3 " 


133-6 


1,540 -^ 


13 


2 





12X12 


T- 3 




2,150 -^ 


14 


2 





12X 12 


I .. 3 " 




2,050 X^ 


15 


9 


9 


12X 12 


r 3 ' 


131-5 


I 118 !- 1^ 


16 


10 





12X12 


I '' 3 " 


136.0 


I ',587 § 


17 


10 





12X 12 


I Portland cernent, 2 sand. 


131 . 


2,003 5 
2,720 


18 


2 


8 


16X 16 


I ," " 2 " 




19 


10 





16X 16 


I' " " 2 " 




i;887jO 


20 


2 





12X12 


I Hme, 3 sand. 




1,370 Bav 


21 


6 





12X 12 


I " 3 " 




1,133 >■ State 


22 


6 





12X12 


I " 3 " 


iig.7 


1,210 bi-icks. 


23* 


6 





12X12 


I lime, 3 sand. 


118. 2 


i,33i1 
1,21 1 


24t 


6 





12X 12 


I " 3 " 


118. 1 


25 

26 


10 


10 



12X12 
12X12 


I 3 ' 


120.3 
118. 


1,174 C/J 

924 -^ 


27 


10 





8X 12 


I 3 " 


107.0 


940 1 -d 


28 


TO 





12X16 


^ " 3 '' 


118. 7 


773 V-^ 


29 


6 





12X 12 


I 3 .1 Rosendale cement. 


I 20. 6 


1,646 ( (U 


30 


6 





12X12 


I Rosendale cement, 2 sand. 


123.0 




1,972 ! 03 ' 


31 


6 





12X12 


I lime, 3 sand, 2 Portland cement. 


I 20 . 3 


1,411 (^ 


32 


6 





12X 12 


I Portland cement, 2 sand. 


119 . 7 


1,792 1 

2,375J 


33 


6 





12X 12 


Clear Portland cement. 


126.6 



* Joints broken every 6 courses. 



t Bricks laid on edge. 



41 8 COMPRESSION. [Ch. VIII. 

were from the Bay State Brick Co., of Boston and Ca.m- 
bridge, Mass., and were medium burnt. 

The brick piers were built of bricks ''laid on beds and 
joints broken every course,, with the exception of two 1 2 by 
1 2 piers, one of which had joints broken every sixth course, 
and one had bricks laid on edge. 

''They were built in the month of May, 1882," and 
"their ages when tested ranged from 14 to 24 months." 
They were all tested between cast-iron plates. 

"Loads were gradually applied in regular increments, 
. . . returning at regular intervals to the initial load. . . . 
Cracks made their appearance at the surfaces of the 
piers and were gradually enlarged before the maximum 
loads were reached. Final failure occurred by the partial 
crushing of some of the bricks, and by the enlargement of 
these cracks, which took a longitudinal direction and 
occurred in the bricks of one course opposite the end joints 
of the bricks in the adjacent courses. This manner oi 
failure was common to all piers. 

It is important to notice that the resistance of the piers 
varies with the strength of the mortar used in the joints. 

Brick piers, 8 inches by 8 inches in cross-section and 
6 feet high, built of Hudson River common brick, and 
others of Sykesville face brick were tested to destruction in 
the testing laboratory of the Department of Civil Engineer- 
ing of Columbia University in 191 5 by Mr. J. S. Macgregor, 
in charge of the laboratory, with the following results, two 
of the piers being built of Hudson River common brick and 
three of the Sykesville face brick. 





Lbs. per sq. in. 




Max. 


Mean. 


Min. 


Hudson River Common 


902 
3,436 


812 
3.363 


722 
3.289 


Sykesville Face ' 





Art. 68.] BRICKS AND BRICK PIERS. 419 

These piers also gave the two following values for the 
modulus of elasticity in compression : 

Hudson River Common E= 748,000 lbs. per sq. in. 

Sykesville Face £^ = 2,860,000 lbs. per sq. in. 

The age of the columns was 60 days. The ends were 
finished with plaster of paris to secure square and uniform 
bearings. The two moduli were determined at intensities 
of stress less than 250 pounds per square inch. 

Mr. Macgregor also obtained the ultimate resistances of 
three piers, 74 inches high built up of single, approximately 
8-inch by 12 -inch hollow tile giving a gross horizontal cross- 
section of, 94 square inches and a net section of actual tile 
material of 50 square inches. 

These tile piers had f-inch joints filled with Portland 
cement mortar, i cement, 3 sand, the age of the piers 
being 60 days. 

The ultimate compressive resistances per square inch 
for the three piers were as follows : 

Gross Section 1,236; 1,239; ^^^ i»ii7 lbs. per sq. in. 

Net Section 2,324; 2,329; and 2,100 lbs. per sq. in. 

These tile piers failed in the blocks in most cases, but 
in other cases in the joints. The failures of the blocks 
showed vertical cracks as well as horizontal and some spalling. 

The results of all the experimental investigations 
available in connection with brick masonry and experiences 
in the best class of engineering work indicate that masonry 
laid up of good hard-burnt common brick may safely 
carry a working load of 15 to 20 tons per square foot or 
210 to 280 pounds per square inch. In the construction 
of this class of masonry where the duties are to be severe it 
is of the utmost importance that the best class of Portland 
cement mortar be employed, as the carrying capacity of 
brick masonry depends largely if not chiefly upon the 
character of the mortar. 



420 COMPRESSION. [Ch. VIII. 

Art. 69. — Natural Building Stones. 

The ultimate compressive resistance of natural building 
stones is affected greatly by the condition of the rock 
from which the cube or other test-piece is taken. That 
portion of a ledge exposed to the weather may be much 
weakened and, in fact, even disintegrated, but the material 
at a short distance from the exterior surface may have the 
greatest resistance of v/hich the particular kind of stone is 
capable of yielding. Again, the compressive resistance 
of stones on their natural beds is much greater than when 
tested on edge. In the tests which follow the test -pieces 
were fairly representative of such quality of stones as 
would pass insfjection in first-class engineering work, and 
it is to be assumed that they were compressed on their 
beds unless otherwise stated. 

Table I taken from the " U. S. Report of Tests of Metals 
and Other Materials " for 1894, exhibits the coefficients of 
elasticity, ultimate compressive resistances, weights per 
cubic foot and coefficients of thermal expansion per degree 
Fahr., as well as the ratio, r, between lateral and direct 
strains for the granites, marbles, limestones, sandstones, 
and other stones shown in the left-hand column. The 
coefficients of elasticity and of thermal expansion were 
determined by employing blocks of stone about 24 ins. 
long and 6 ins. by 4 ins. in cross-section, the gauged length 
being 20 inches, but the ultimate compressive resistances 
were found by testing 4-inch, cubes. The number of tests 
for each coefficient of elasticit}^ and ultimate resistance 
varied from one to nine but were generally two or three. 
The general run of values of ultimate resistance will be 
found to conform as well as could be expected with results 
for the same kind of stones in the tables which follow. 



Art. 69.] 



NATURAL BUILDING STONES. 



421 



It will be observed that the marbles are the heaviest stones, 
although the granites are not much lighter. There is a 
large difference, however, between the sandstones and the 
marbles or granites. 

Table I. 

NATURAL STONES IN COMPRESSION ON BEDS. 



Stone. 


Coefficient 

of 

Elasticity, 

Lbs. per 

Sq. In. 


Ultimate Compressive 

Resistance, Lbs. per 

Sq. In. 


Weight 

per 

Cu. Ft., 

Lbs. 


Coefficient 
of Expan- 
sion per 
Degree 

Fahr. 


r. 




Max. 


Mean. 


Min. 




Branford granite, Conn .... 


8,712,100 


15,854 


15,707 


15,560 


162 


.00000398 


I 
4 


Milford granite. Mass 


7,676,750 


25,738 


23,773 


19,258 


162.5 


.00000418 


I 
5.8 


Troy granite, N. H 


6,118,850 


28,768 


26,174 


23,580 


164.7 


.00000337 


r 
5-1 


Milford pink granite, Mass. . 
Pigeon Hill granite, Mass. . . 
Creole marble Ga. ... 


6,200,350 
8,095,250 
7,993,-00 

10 427,800 


22,162 

20,716 

I5,5T2 

13,4^5 


18,988 
19,670 
13,466 

12,619 


15,756 
17,772 
1 1,420 

11,822 


161. 9 
161. 5 
170 

167.8 










I 


Cherokee marble, Ga 


. 00000441 


2.9 

r 

S.7 


Etowah marble, Ga 


8,792,600 


14,217 


14,053 


13,888 


169.8 




I 




^..6 


Kennesaw marble, Ga 


8,217,950 


10,771 


9,563 


8,354 


168. I 




I 




3.9 


T V,! AT 










168.6 


.00000454 
. 00000202 


Marble Hill marble, Ga. . . . 


9,950,850 


11,532 


11,505 


11,478 


I 
S.4 


Tuckhoe marble, N. Y 


IS 173,200 


19,223 


16,203 


11,640 


178 


.00000441 


4- 5 


Mount Vernon limestone, Ky 


3,278,400 


11,566 


7,647 


5,247 


I39-I 


.00000464 


4 


Oolitic limestone, Ind 

North River bluestone, N. Y. 

Manson slate, Maine 

Cooper sandstone, Oregon .. . 


5,475,300 


EE 




~zz 





.00000437 
.00000519 


22,947 
14,920 




















Cooper sandstone, Oregon . . 


3,021,350 


16,366 


15,284 


14,203 


159-8 


.00000177 


II 


Maynard sandstone, Mass. . . 


2,034,650 


10,538 


9,880 


9,223 


133-5 


.00000567 


- 


Kibbe sandstone, Mass 


2,066,800 


10,663 


10,363 


10,063 


133-4 


.00000577 


S- ^ 


Worcester sandstone, Mass. . 


2,668,750 


9,869 


9,763 


9,656 


136.6 


.00000517 


4.4 


Potomac sandstone, Md. . . 
Olvmpia sandstone, Oregon. 
Chuckanut sandstone. Wash. 

DvckerhofF's cement 

* Yammerthal flint lime- 




13,441 
12,790 


T 2,665 
11,389 

23,724 
28,647 


I 2,061 
10,276 

18,496 




. 000005 
. 0000032 















.00000578 






28,951 



























* From Report of 



42 2 COMPRESSION. fCh. VIII. 

The coefficients of elasticity generally range considerably 
higher than those for concrete in Art. 67, but the sand- 
stones form an exception to this observation. The coeffi- 
cients of thermal expansion vary between rather wide 
limits but they are mostly a little lower only than those 
determined for concrete. The coefficient for the Dycker- 
hoff cement is very close to those exhibited for cement 
mortar and concrete in Art. 60. The column headed r, 
giving the ratios between lateral and direct strains, contains 
interesting data. From what has been shown in Art. 4 
it is apparent that the total volume of the test-pieces was 
considerably reduced by the compression to which the 
cubes were subjected. 

The coefficients of elasticity were determined at in- 
tensities of pressure running from 1000 or 2000 pounds 
per square inch up to 8000 or 10,000 pounds per square 
inch. 

A coefficient would first be determined at comparatively 
low pressures, as from 1000 to 3000 pounds per square 
inch, and then at higher pressures, as from 7000 to 9000 
or 10,000 pounds per square inch. As a rule, the co- 
efficients determined at the higher pressures were mate- 
rially higher in value than the others, the stiffness of the 
stone increasing with the loads within the limits of the test. 
The values in the table are the m.eans of those at the low 
and high pressures. 

With the ordinary working values of pressures in 
masonry, probably not more than two thirds of the 
values of the coefficients of elasticity given in the table 
should be employed. 

In the *' U. S. Report of Tests of Metals and Other Mate- 
rials " for 1900 there may be found the results of compress- 
ing 4-inch cubes of Tennessee marble and of granite from 
the Mount Waldo Quarries at Frankfort, Llaine. The 



Art. 69.] NATURAL BUILDING STONES. 423 

ultimate compressive resistances of the 4-inch Tennessee 
marble cubes expressed in pounds per square inch, were as 
follows : 

Maximum. Mean. Minimum. 

25,478 20,329 16,309 

The preceding three results cover twenty tests. 

The ultimate resistances in pounds per square inch 
of the "Black Granite" from the Waldo Quarries, as 
determined from four tests of 2 -inch cubes, were as follows: 

Maximum. Mean. Minimum. 

32,635 30.949 29,183 

Again, in the same report, the ultimate resistances in 
pounds per square inch of four 4-inch cubes of limestone 
from Carthage, Mo., are as follows* 

Maximum. Mean. Minimum. 

17,130 14,947 13.660 

The preceding tests and the results of others given in 
Table II have been determined by compressing cubes 4 
inches and 5 inches on the edge and it has been generally 
customary to use a cube for a test piece for either natural 
or artificial stones. It has already been indicated, however, 
in Art. 62 that such a short test piece in compression must 
necessarily give higher results than should be credited to 
the material. 

The use of compressive test specimens with lengths 
two to two and one-half times the diameter is just begin- 
ning, but that use has not become sufficiently general, nor 
has it been long enough the practice, to make available 
results from such desirable tests. 

Furthermore, some tests have shown that ultimate com- 
pressive resistances may be materially higher for large cubes 



424 



COMPRESSION. 



[Ch. VIII. 



than for small ones. This is probably due to the lateral 
supporting effect given to parts of the test piece by the 
friction between the bearing head of the riiachine and the 
face of the material under test with which it is in contact. 
Preferably no cube tested for engineering purposes should 
be less than 12 by 12 inches in section, nor should any test 
piece be shorter than twice its diameter. 

The results found in Table II are taken from the '' U. S. 
Report of Tests of Metals and Other Materials," for 1894. 
They relate to the various kinds of rock indicated and 
were found by testing 4-inch to 5 -inch cubes on their beds. 

Table II. 



state. 


Stone. 


Ultimate 
Compressive 
Resistance, 
Pounds per 
Square Inch. 


Minnesota 


Ortonville granite. 

Kasota pink limestone 


20,415 
10,833 
17,780 

4,353 
9,606 

8,775 

10,114 

21,556 

19,875 

9,465 

4,834 

2,899 


«. 


Faribault marble 


tc 


Duluth brownstone 


a 


Mankato sandstone 


<< 


IMantorville sandstone 


(( 


Frontinac sandstone 


<< 


Luverne ciuartzite 


<< 




Iowa 


Rubble rock 




Firestone 


<< 


Gypsum Fort Dodge 







The ultimate resistances of the sandstones are relatively 
low, while the higher values are found for granites, lime- 
stones, and quartzites, as is usual. 

In 1906 the Carnegie Institution of Washington pub- 
lished An Investigation into the Elastic Constants of Rocks, 
More Especially with Reference to Cubic Compressibility, 
by Mr. Frank D. Adams and Dr. Ernest G. Coker. The 
experimental part of this investigation was made atMcGill 
University under the auspices of the Carnegie Institution. 



Art. 69.] 



NATURAL BUILDING STONES. 



425 



Although this investigation was made as a contribution 
more to physics than to engineering, the results obtained 
are of both interest and value to engineers and it is well 
to make use even for engineering purpOvSes of results deter- 
mined with so much care and such extreme accuracy in 
vspite of the fact that the specimens used were only i inch 
square in section or i inch in diameter and 3 inches long. 
If E is the ordinary modulus of elasticity in compression 
G the modulus of elasticity for shearing, V the so-called 
bulk. modulus, i.e., the reciprocal of the rate of change of 
unit volume for unit intensity of stress, and r the ratio of 
the rate of lateral strain of the specimen divided by the 
rate of direct strain under compression, Table III gives 
the results of these experimental determinations for those 
materials which American engineers more commonly use. 

Table III. 



Specimen. 


E. 


r. 


G. 


V- ^• 

3(1 -2r) 


Black Belgian marble . 


11,070,000 


0.2780 


4,330,000 


8,303,000 


Carrara marble 


8,04.6,000 


0.2744 


3,154,000 


5,946,000 


Vermont marble 


7,592,000 


0.2630 


3,000,000 


5,341,000 


Tennessee marble 


9,006,000 


0.2513 


3,607,000 


5,967,000 


Montreal limestone . . . 


9,205,000 


0.2522 


3,636,000 


6,167,500 


Baveno granite 


6,833,000 


0.2528 


2,724,800 


4,604,000 


Peterhead granite .... 


8,295,000 


0.2II2 


3,399,000 


4,792,000 


Lily Lake granite 


8,165,000 


0. 1982 


3,380,000 


4.517,500 


Westerly granite 


7,394.500 


0.2195 


3.019.700 


4,397.500 


Quincy granite ( i ) . . . . 


6,747,000 


0.2152 


2,781,600 


3,984,000 


Quincy granite (2). . . . 


8.. 247, 500 


0.1977 


3,445 000 


4.-555.000 


Stanstead granite 


5,685,000 


0.2585 


2,258,700 


3,940,000 


Ohio sandstone 


2,290,000 


0.2900 


888,000 


i.8i6,oo« 


Plate glass 


10,500,000 


0.2273 


4,290,000 


6,448,000 





426 COMPRESSION. [Ch. VIII. 

Art. 70. — Timber. 

The ultimate compressive resistance, coefficient of elas- 
ticity, and other physical properties of timber in com- 
pression are affected greatly by the amount of moisture 
in the timber and by the size of stick. The investigations 
of Professor J. B. Johnson, acting for the Forestry Division 
of the U. S. Department of Agriculture, have shown that 
when the amount of moisture exceeds about 30% by 
weight of the timber the physical properties are not m.uch 
affected by any increased saturation. The walls of the 
wood cells at that point seem to experience their maximum 
softening. Green timber may be considered as carrying 
about one third of its weight in moisture, and it seems to 
matter little whether that moisture is water or sap, timber 
once dried and resaturated appearing to suffer the same 
diminished resistance as in its original green condition. 
Professor Johnson's tests showed that the Southern pines 
increased their ultimate compressive resistance in some 
cases as much as 75% by the process of drying or seasoning 
from 33% of moisture down to 10%, the general rule being 
a greatly increased compressive resistance with a decrease of 
moisture. It follows from these results, therefore, that green 
timber will be much weaker in compression than seasoned 
timber. Ordinary air seasoning even under cover seldom 
reduces moisture below about 15% in w^eight of the timber 
itself, although under favorable circumstances of seasoning 
the moisture may sometimes drop to 12% of that w^eight. 
As a matter of precision, therefore, or accuracy, the ulti- 
mate compressive resistance of timber should always 
be stated in connection with the percentage of moisture 
carried by the timber. This will be found to be the case 
in all of Professor Johnson's experimental work, to which 
reference has already been made and the results of which 



Art. 70.1 TIMBER. 427 

are chiefly found in bulletins Nos. 8 and 1 5 of the Division 
of Forestry of the U. S. Department of Agriculture, the 
former being dated 1893. 

The earlier tests of Professor Johnson were made on a 
basis of 15% moisture, but in his later work a basis of 12% 
moisture was adopted, and he states in Circular No. 15 
that in reducing the moisture from 15% to 12% the corre- 
sponding increases in the ultimate compressive resistance 
in pounds per square inch of Southern pines are approxi- 
mately as follows: 



Endwise. 



Across Grain. 



Long-leaf pine . 
Cuban pine. . . . 
Loblolly pine. . 
Short-leaf pine , 



1,100 
800 
900 
600 



180 

220 

150 

60 



While it is important as a matter of physics to recognize 
clearly the effect of moisture upon the compressive re- 
sistance of timber, it is of equal importance, and possibly 
of greater importance, to recognize the fact that in engineer- 
ing practice, except in specially protected cases, the timber 
used in structures is more or less exposed and can seldom 
or never be depended upon to contain even as little as 15% 
of moisture, and with some conditions of weather and at 
some seasons of the year it may contain considerably more. 
It follows, also, that the condition of timber as to moisture 
in most structures will change materially from time to time. 
It would be unwise, therefore, and perhaps dangerous to use 
working compressive resistances based upon the results of 
tests of small pieces with moisture reduced to 15% or 12%, 

i\gain, it has been frequently stated as a result of the 
timber investigations by the Forestry Division of the 
U. S. Department of Agriculture, that the ultimate com- 



4-'8 COMPRESSION. [Chi VII I. 

pressive resistance of large sticks may be taken as practically 
identical with that belonging to small selected test pieces, 
the quality of the material being the same in both cases. 
It is possible, if the quality of material throughout all 
portions of every large stick were identical with the quality 
of small selected specimens, that the ultimate compressive 
resistance per square inch might be the same; but that is 
radically different from the facts as they are. There is 
probably no stick of timber whose condition is permanent 
at any given time. If it is seasoning, its qualit}^ is im- 
proving, but after reaching a maximum of excellence it 
begins to depreciate by decay or from other causes. Any 
large stick of timber as used by the engineer is seldom 
free from some point of incipient decay and it is never 
free from knots, large or small, wind shakes, cracks from 
one catise or another, or from some other defective con- 
dition, at some point. Small specimens for testing are 
invariably so selected as to eliminate such spots as militating 
against a comparatively high resistance. The inevitable 
result for full-size sticks is a decreased resistance materially 
below that of the small specimen. For all these reasons, 
therefore, in engineering practice it would be a radical 
error to accept the ultimate compressive resistance per 
square inch of small test specimens as practically identical 
with that of large sticks. Values for the latter class of 
timber should be determined upon pieces as large as those 
used in structures and under the same conditions in which 
they are used, which means an indefinite amount of moisture 
ordinarily sensibly larger than 12% or 15%. 

In the "U. S. Report of Tests of Metals and Other 
Materials" for 1896 and 1897 there may be found results 
of compressive tests for coefficients of elasticity for sticks of 
timber as shown in Table I. Those sticks were many of 
them large enough to form full-size posts. They appear to 




The fracture of a piece of Douglass fir or Oregon pine loaded tangentially to 
the rings of growth. The ultimate compressive resistance was found to be 600 
lbs. per sq. in. 

iTo face page 429.) 



Art. 70.] 



TIMBER. 



429 



Table I. 
TIMBER IN COMPRESSION. 



Kind of Wood. 


Coefficient of Elasticity, 
Pounds per Square Inch. 


1 

6 


Remarks. 




Maximum. 


Mean. 


Minimum. 




Douglas fir : 

Endwise 


3,461,000 
112,000 
207,000 

1,789,000 

1,890,000 
2,252,000 

1,655,000 

1,623,000 

2,300,000 


2,358,000 

74,600 

158,000 

1,554,000 

1,657,000 
2,175,000 

1,469,500 

1,531,000 

2,251,000 


1,915,000 

40,000 

134,300 

1,338,000 

1,488,000 
2,049,000 

1,202,000 

1,437,000 

2,207,000 


4 
9 
6 

6 

4 
-1 

6 
10 

12 


Not well seasoned. 


Tangentially 

Radially 




White oak : 
Endwise 


(( (( (( 


Long-leaf pine : 

Endwise . . . 


From tops of trees 
From butts of trees 

Not well seasoned 


Short-leaf pine:* 
Endwise . . , . . . . 


Spruce : * 

Endwise 


(( (t (I 


Old yellow-pine posts :* 
Endwise 


Very dry. 





* These results are means of determinations at intensities varying from 
500 to 5,000 pounds per square inch. 

have been of merchantable timber of about such quahty as is 
used in first-class engineering works. They had the usual 
supply of knots and other features which, while not material 
defects, prevented the pieces from being of selected quality. 
As also shown in the table, there were a considerable 
number of tests in each case. • "Endwise" compres- 
sion means compression parallel to the fibres of the 
timber, while "Tangentially" means a direction tangent to 
the rings of growth. That compression indicated by 
"Radially" was in a radial direction, i.e., passing through 
the centre of the tree trunk. The determinations were 
made at intensities of pressure varying from one third to 
one half the ultimate resistance. It will be noticed that 
in the values for long-leaf pine the highest results belong 
to sticks from the butts of trees, while those from the tops 



430 COMPRESSION. [Ch. VIIL 

give materially less values. It will also be observed that 
the values for the very dry yellow-pine posts in the last 
line of the table are high, showing the increased stiffness 
due to the absence of moisture. The coefficients of elas- 
ticity in the last five lines of the table were computed 
from the resilience of the compressed columns by means 
of a formula similar to eq. (2) of Art. 44. 

The values of the elastic limit, ultimate resistance and 
modulus of elasticity in compression along the fibres as 
well as the elastic limit in compression across the fibres of 
nine of the prominent structural timbers of the United 
States, both for large or structural sizes and small speci- 
mens, as shown in Table II, are taken from Tests of Struc- 
tural Timbers, Forest Service-Bulletin 108, U. S. Depart- 
ment of Agriculture, by Messrs. McGarvey Cline and A. L. 
Heim, 191 2, and exhibit some of the latest experimental 
investigations in the elasticity and resistance of timber. 
The large or structural sizes had cross-sections up to 10 
inches by 16 inches and the small sizes down to 2 inches 
by 2 inches. The resistances parallel to the fibres, i.e. on 
end, were determined for pieces whose lengths were three 
to four times the cross dimensions. 

The authors of the paper properly observe that the 
" Results of tests made only on small thoroughly seasoned 
specimens free from defects " — " may be from one and one- 
half to two times as high as stresses developed in large 
timbers and joists." This is an important conclusion and 
a number of results in Table II confirm the observations 
of the authors. 

It is essential to observe the small resisting capacity 
of the various timbers when compressed across the grain, 
the resistance in the latter condition being but a small 
fraction of that along the grain. 

Table III contains the results of tests by Colonel Laidley, 



Art. 70.] 



TIMBER. 



431 



tn.t; ft c 



5 .^ 
<u 2 5 






J5g 



00 -r)- 

COOO 

o o 



\D vC 


,^ 


10 


• l^ 


ON C 


ON 


10 


• ON 


t^ OS «0 


vO 


• MD 












: 


Q 






Q 















^ 






VO 


i_i 


10 









10 : 


ON 






(N 


rO . 












*~' 






'"' 


'"' 



88 

q q_ 

O' CO 

•"1. "^ 



00 CM 

10 o 

c< o 



O O 
rooo 
O CO 



Th CO 
r^ On 



ON '^ 
M 10 



o o 

CS OS 



^ CO 
h-, O 
00 "^ 



■^10 \0 M3 



0< 00 



o o 
00 »o 



00 o 

04 10 
04 10 



W^^Oh' 



hi o 
10 O 
CO ^ 



to 



'^ ON 

COVO 



< 






S 2 e 






O O 
O O 

9- 9. 
tF 10 

M O) 

Ti-Os 



o o 
o o 
9. 9 

10 10 
10 10 



88 

c o__ 

w CO 

vo r^ 



o o 

o 
9 9 
o" oT 
"d-o^ 

01 01 



O O 

o o 
00 ri- 



10 o 

C> CO 
^ O 



10 o 
CO t^ 



O VO 

M ON 

10 vO 



o o 

On 01 



o o 

CO ON 
01 HH 



»0 01 
10 ON 

CO CO 



01 O 

00 00 
00 On 



O 
00 
'^ 
CO 



o o 
l-^ 10 



10 vO 

t^ 01 

MD O 



1000 

O CO 

On on 



ON ON 



10 'd- 
10 c 
10 10 



ft3j 
O 



c 


a; 


in 

(1) 


a 




N 














CTi 




< — 1 


cu 


ox 


rT, 


Ml Gj 


P 





h4c/2 







"^ W (-! ^J vj ^ , w>> lyi 

^- ,N N qL,.^ N j^ '-■' -^ 

M=l 'w "w <^ "w "55 -—I 



c« 


r/5 


O) W 


w w 


V2 C 
G 


m 


w 


tn 


r^ 


5 fN> 


rn 




0) 




(U (U 






<X> 


0) 




N 






tS) •• N 


N CU 


N 




N 


N 




w 


■55 


U-55 


•55r^-w 


•oc-^ 


C/J 


w^ 


i'^ 


'w 


a'w 


■55 



Q CO ^ h^ H ^ P< 






43 2 



COMPRESSION. 
Table III. 



[Ch. VI ri. 



No. 



Kind of Wood. 



Oregon pine , 

Oregon pine 

Oregon pine 

Oregon maple 

Oregon spruce 

California laurel 

Ava Mexicana 

Oregon ash 

Mexican white mahogany 

Mexican cedar 

Mexican mahogany 

White maple 

White maple 

Red birch 

Red birch 

Whitewood 

Whitewood 

White pine 

White pine 

White oak 

White oak 

Ash 

Ash 

Oregon pine 

Oregon maple 

Oregon spruce 

Oregon spruce 

California laurel 

Ava Mexicana 

Oregon ash 

Mexican white mahogany 

Mexican cedar 

Mexican mahogany 

White pine 

White pine 

Whitewood 

Whitewood 

Black walnut 

Black walnut 

Black walnut 

White oak 

Spruce 

Yellow pine 

Black walnut 

Black walnut 

Black walnut 

Black walnut 

Black walnut 

Black walnut 

White pine 

White pine 

White pine 

White pine 

White pine 

White pine 

Yellow birch 

Yellow birch 

White maple 

White maple 

White oak ' . . 



Length, 
Inches. 



16. 5 

[Q.Q 



12. O 
12. O 

1-95 

3.63 

3-92 

3.92 

3.58 

3-69 

3-64 

3-77 

3-75 

3-75 

3 -06 

2.90 

3-15 

3.15 

0-875 

0.875 

0.875 

2 . 40 

3.70 

3.90 

0-75 

1 . 00 

1-25 

1.50 

1-75 

2 .06 
0.7S 

1 . 00 

1-25 

1.50 

1-75 

2 .00 
4-25 
4-25 
4. 00 
4. 00 
3-95 



Compressed 
Section 

in 
Inches. 



2. 46 X 2.0 
1.21X 1.21 
1 . 21 X 1 . 21 
3-63X3.63 
3-92X5.75 
3.58X3-58 
3.69X3.69 
3.64X3.64 
3.77X3.77 
3-75X3.75 
3.75X3.75 
4.00X4.00 
4.00X4.00 
4.26X4.26 
4.26X4.26 
4.00X4.00 
4.00X4.00 
4.00X4.00 
4- 00 X 4. 00 
4.00X4. 00 
4.00 X 4. 00 
4- 00 X 4- 00 
4.00X4.00 
3.45X3.00 
3.63X3.00 
5.75X4.75 
4.75X4.-00 
3.58X3.00 
3.69X3.00 
3.64X3.00 
3.77X3.00 
3.75X3.00 
3.75X3.00 
20X4.75 



4-75X4.00 
4-75X6.20 
4.75X4.00 
4-75X4.00 
4 . 00 X 3 • 94 
4.00X 2.50 
4.75X4.00 
4.75X4.00 
4.00X4.00 
4.05X4.00 
4.05X4.00 
4.05X4.00 
4.05X4.00 
4.05X4.00 
4.05 X4. 00 
4.05X4.00 
4.05X4.00 
4.05 X4. 00 
4.05X4.00 
4.05X4.00 
4.05X4.00 
4. 25X3.00 
5.98X3.00 
3.95X3.00 
5 . 98 X 3 . 00 
3 .96X 3-00 



Ultiinate 
Resist- 
ance, 
Pounds 

per 
Square 
Inch. 



8,496 
9,041 
8253 
6,661 

5,772 
6,734 
6,382 
5,121 
6,155 
4,814 
[0,043 
7,140 
7,210 
8,030 
7,820 
4,440 
4,330 
5,475 
5,760 
7,375 
7.010 
7,940 
7,640 
1,150 
1,875 

710 

680 
2,000 
2,100 
2,200 
2,150 
1,950 
4,500 

875 
1,012 

900 
1,000 
2,450 
2,200 
2,525 
3,550 

970 
1,900 
2,800 
2,56c 
2,400 
2,500 
2,400 
2,360 
1 ,1 20 
1 ,100 
1,160 
1,070 
1,060 
1,000 
2,000 
1,650 
1,700 
1,900 

2,sOO 



C r t 

(LI Cq 



With 



Remarks. 



Unseasoned 
Worm-eaten 



Unseasoned 
Unseasoned 



Mean of two 
Mean of two 
Mean of foiir 
Mean of two 
Mean of two 
Mean of two 

Mean of foiir 



Mean of two 



Art. 70.] TIMBER. 433 

U.S.A., "Ex. Doc. No. 12, 47th Congress, 2d Session." 
A few other tests of short blocks from the same source will 
be found in the article on "Timber Columns." Unless 
otherwise stated, all the specimens were thoroughly sea- 
soned. ■ 

, In this table the "length" of all those pieces which 
were compressed in a direction perpendicular to the grain 
might, with greater propriety, be called the thickness, since 
it is measured across the grain. 

In the tests (24-60) the compressing force was dis- 
tributed over only a portion of the face of the block on 
which it was applied ; thus the compressed area was sup- 
ported, on the face of application, by material about it 
carrying no pressure. In some cases this rectangular com- 
pressed area extended across the block in one direction, 
but not in the other. In all such instances the ultimate 
resistance w^as a little less than in those in which the area 
of compression was supported on all its sides. 

The "ultimate resistance" was taken to be that pressure 
which caused an indentation of 0.05 inch. 

Nos. (44-5 5 ) show the effect of varying thickness of blocks. 
Within the limits of the experiments, the ultimate resistance 
is seen to decrease somewhat as the thickness increases. 

The best series of values of the ultimate compressive re- 
sistance of timbers as actually used in large pieces and for 
engineering structures that can be written at the present 
time is that given in Table IV. 

That table shows values for railway bridges and trestles 
adopted by the American Railway Engineering Associa- 
tion. 

As in the case of tension, the compressive resistances across 
the grain are but small fractions of those with the grain. 
Values are given for columns under 15 diameters in length 
for the reason that such columns fail essentially by com- 



434 



COMPRESSION. 



[Ch. VIII. 



pression and without the bending which characterizes long 
columns. The table is one of great practical value. 



Table IV. 

TIMBER IN COMPRESSION. 





Unit Stresses in Lbs. per Sq. In. 


Kind of Timber. 


Perpendicular to 
the Grain. 


Parallel to 
the Grain. 


Working Stresses 
for Columns. 




Elastic 
Limit. 


Working 
Stress. 


Mean 

Ult. 


Working 

Stress. 


Length 
Under 
15 Xd. 


Length Over 15 Xd. 


Douglas fir ... . 
Longleaf pine . . 
Shortleaf pine. . 
White pine .... 

Spruce 

Norway pine. . . 

Tamarack 

Western Hem- 
lock 

Redwood 

Bald cypress. . . 

Red cedar 

White oak 


630 
520 
340 
290 
370 

440 
400 
340 
470 
920 


310 
260 
170 
150 
180 
150 
220 

220 
150 
170 
230 
450 


3,600 

3.800 

3.400 

3.000 

3.200 

2,600* 

3.200* 

3.500 
3.300 
3.900 
2,800 
3.500 


1,200 
1,300 
1,100 
I.OOO 

1,100 

800 
1,000 

1,200 
900 

1,100 
900 

1,300 


900 

975 
825 
750 
825 
600 
750 

900 
675 
825 
675 
975 


1,200(1 -//6orf) 
1,300(1 -//6orf) 
1,100(1— //6o</) 
1,000(1 -//6o4 
1,100(1 -//6o^) 
800(1 -//60J) 
1,000(1— //6o^) 

1.200(1— //6o<f) 
900(1 -l/6od) 

1,100(1 -//6o^) 
900(1— //6orf) 

1,300(1 -//6oc?) 



Unit stresses are for green timber and are to be used without increasing 
the live load stresses for impact. Values noted * are partially air-dry 
timbers. 

In the formulas given for columns, /= length of column, in inches, and 
d = least side or diameter, in inches. 



. CHAPTER IX. 

RIVETED JOINTS AND PIN CONNECTION. 

Art. 71. — Riveted Joints. 

Although riveted joints possess certain characteristics 
under all circumstances, yet those adapted to boiler and 
similar work differ to some extent from those found in the 
best riveted trusses. The former must be steam- and water- 
tight, while such considerations do not influence the design 
of the latter, consequently far greater pitch may be found 
in riveted-truss work than in boilers. Again, the peculiar 
requirements of bridge and roof work frequently demand 
a greater overlap at joints and different distribution of 
rivets than would be permissible in boilers. 

. Kinds of Joints. 

Some of the principal kinds of joints are shown in Figs, i 
to 6. Fig. I is a "lap-joint" single-riveted; Fig. 2 is a 
"lap-joint" double-riveted; Fig. 3 is a "butt-joint" with 
a single butt-strap and single-riveted; whi^e Figs. 4, 5, and 
9 are "butt-joints" with double butt-straps. Fig. 4 being 
single-riveted, while the others are double-riveted. Fig. 5 
shows zigzag riveting, and Fig. 6 chain riveting. All these 
joints are designed to resist tension and to convey stress 
from one single thickness of plate to another. Two or 

435 



436 




RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

n 



OOO 
OOO 



OOO 



OOO 



} 





OOO 

ooo 





I^^G. I. 



Fig. 2. 



) o o c 
ooo 



c 

o o o c 
b o o cc 



Fig. 


3- 




o o 
oo 


o 
o 


ooo 
ooo 





If 



Fig. 4. 



Fig. 5. 



Fig. 6. 



G-P--0) O 

e^o o'o'o 

O -i-Q-'- O 



r i 



fth 



■p 



:) 



:) B 



^ 



T- 



2 



:) 




Fig. 7. 



Fig. 



Art. 72.] DISTRIBUTION OF STRESS IN RIVETED JOINTS. 437 

three other joints peculiar to bridge and roof work will 
hereafter be shown. 

In the cases of bridges and roofs these ''butt-straps" 
are usually called '^ cover-plates." 

Art, 72. — Distribution of Stress in Riveted Joints. 

Bending of the Plates. 

In order that rivets, butt-straps or cover-plates and 
different parts of the main plates may take their proper 
stresses, an accurate adjustment of these different parts to 
the external forces or loads must be attained ; but all shop 
work is necessarily more or less imperfect and the varying 
stresses at different parts of the joint produce at least 
elastic deformations so that the requisite conditions for a 
proper distribution of stresses as computed cannot be main- 
tained. The precise amount of stress, therefore, carried 
by each rivet, cover-plate or other part of the joint in- 
cluding the main plates cannot be computed. By means 
of reasonable assumptions, however, and by the introduc- 
tion of factors or coefficients determined by the actual 
testing of riveted joints, simple and sufficiently accurate 
formulae for all engineering purposes may be established. 

The shafts of the rivets of any joint compress or 
bear against the walls of the rivet holes in the transference 
of loading from one main plate to the other. This con- 
dition will necessarily subject the metal on either side of 
the hole and adjacent to it to a higher degree of tension 
than the metal midway between two neighboring holes. 
This makes the average intensity of stress over the minimum 
section of either the main plate or the cover-plate materi- 
ally less than the maximum intensity at or near the wall 
of the rivet hole. On the other hand, the removal of the 
metal for the rivet holes makes that part of the plates 



438 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

between two consecutive holes at right-angles to the direc- 
tion of loading a " short " specimen with a higher ultimate 
resistance than a long specimen. 

Again let Fig. 8, like Fig. 2 of the preceding article, 
represent a longitudinal section of a double riveted lap- 
joint, the thicknesses of the two plates being t and f. The 
two opposite loads P would produce a bending moment 
about an axis at right-angles to the plane of section of 

P . Usually the two thicknesses of plate are the same 

2 

making t the lever arm of the couple. This moment causes 
bending in the plates in the vicinity of A and B of equal 
amount and the bending intensities of stresses may be com- 
puted in the usual manner if the joints were not distorted 
so as to change the lever arm of the couple. As the load 
is increased, however, the joint tends to take the shape 
shown in Fig. 9, the two plates tending to pull into the 
same straight line, making it impossible to compute accu- 
rately the bending moment. It is sufficient, however, to 
recognize this condition of flexure in the joint. 

This eccentric action of the load P produces also the 
same bending moment in the rivets of the joint, in the 
aggregate, as that impressed upon the plates. The assumed 
bending moment carried by each rivet will be the moment 

t-\-f 
P — — or Pt divided by the number of rivets in the joint. 
2 

This bending moment is seldom or never computed for 
rivets but it is always computed in the design of pins of 
a pin-connected truss bridge. 

For all these reasons and others shortly to be considered 
it is obvious that if a riveted joint of any type be tested to 
destruction, it is essentially impossible to compute accu- 
rately what the intensity of stress will be in any part of 
it at any stage of loading. Such tests, however, yield most 



Art. ']2\ DISTRIBUTION OF STRESS IN RIVETED JOINTS. 439 

valuably empirical quantities to be used in formulas to be 
established and without which it would be essentially 
impossible to design a riveted joint in a rational manner. 

Although these considerations are based upon the charac- 
teristics of a double-riveted lap-joint, they apply to all 
riveted joints of any type whatever. If the butt-joint with 
double cover-plates shown in Fig. 5 of the preceding 
article be considered, it will be clear at once that if a line 
be drawn centrally through the section of the two main 
plates, each half of the actual joint will be divided into 
two equal double-riveted lap-joints in each of which the 
plates will, be subjected at least approximately to the same 
condition of stress as that found in connection with Fig. 9 
and the bending of the rivets will be precisely the same. 
There will be, however, no bending of the main plates. 

The special form of joint shown in Fig. 7, which has 
come to be much used, will also have its parts subjected 
to the same general condition of stresses including the bend- 
ing of rivets and main plates. 

It is clear that the bending of the plates illustrated in 
Fig. 9 will increase with their thickness. 

Net Section of Plates 

The net section of any main plate or cover-plate in a 
riveted joint is the gross section along any transverse line 
of rivets less the metal taken out by the rivet holes. In 
Fig. 2 of the preceding article, the net section of either main 
plate will be its gross section less three rivet holes. The 
pitch p of the rivets in any transverse line of rivet holes 
in a riveted joint is the distance between the centres of two 
consecutive rivets as shown in Fig. 7. In the centre line 
of rivets in that figure, the pitch is one-half that in the outer 
line. The net section of any plate, therefore, per rivet will 



440 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

be (p-d)t, d being the diameter of the rivet hole and t 
the thickness of the plate. If n is the number of rivets 
in one main plate and if q is the number of rows of rivets 

in it, then the number of rivets in each row will be - and the 

q. 

total net section along any transverse row of rivets will 
be-{p-d)t. 

Bending of the Rivets. 

It has already been seen that the rivets of any riveted 

joint are subjected to bending. It is assumed that the 

t + t' 
total bending moment, M=P or M=Pt is divided 

2 

uniformly among all the rivets of the joint. Hence the 
bending moment to which a single rivet is subjected is 

M kAd , V 

— =-—-. ....... (i) 

n o 



in which A is the area of cross section of one rivet 
and k the greatest intensity of tension or compression in 
the extrem^e fibre due to bending. By introducing in eq. 
(i) the values of M already used, eqs. (2) and (3) at once 
result. 



iit=i', 



^=«^^^^ (3) 



This equation is approximate because it is virtually 
assumed that the pressure on the rivet is uniformly dis- 



Art. 72.] DISTRIBUTION OF STRESS IN RIVETED JOINTS. 441 

tributed along its axis.* This is a considerable deviation 
from the truth, particularly as failure is approached. The 
true bending moment is much less than that given by 
eq. (i) after the rivet has deflected a little. 

When the joint takes the position shown in Fig. 9, it 
is clear that the rivet is also subject to some direct tension. 

The Bearing Capacity of Rivets. 

There is a very high intensity of pressure between the 
shaft of the rivet and the wall of the hole. This intensity 
is not uniform over the surface of contact, but has its 
greatest value at, or in the vicinity of, the extremities of 
that diameter lying in the direction of the stress exerted 
in the plate. At and near failure this intensity may be 
equal to the crushing resistance of the material over a con- 
siderable portion of the surface of contact. 

The intricate character of the conditions involved ren- 
ders it quite impossib^^e to determine the law of the dis- 
tribution of this pressure. The bending of the rivets under 
stress tends to a concentration of the pressure near the 
surface of contact of the joined plates, while the unavoid- 
ably varying "fit" of the rivet in its hole, even in the best 
of work, throws the pressure towards the front portion of 
the surface of the rivet shaft. The intensity thus varies 
both along the axis and around the circumference of the 
rivet. 

If any arbitrary law is assumed, the greatest intensity 
of pressure is easily determined. Such laws, however, are 
mere hypotheses and possess no real value. All that can 
be done is to determine, by experiment, the mean safe 

* In accordance with this assumption, strictly speaking, ^t (thickness of 
main plate) should be taken instead of t in the sum (/+/') in the above 
formulae for bending, when appHed to the double butt-joint, Figs. 5 and 6. 



442 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

working intensity on the diametral plane of the rivet which 
is equivalent to a fluid pressure of the same intensity against 
its shaft. 

Thus, if / is this mean (empirically determined) intensity, 
d the diameter of the rivet, and t the thickness of the plate, 
the total pressure carried by one rivet pressing against one 
plate is 

R=fdt (4) 

Bending of Plate Metal in Front of Rivets. 

In addition to the bending of the plates of a riveted joint 
about an axis parallel to the plates and at right angles to 
the direction of loading, there is further bending of the 
metal immediately in front of a rivet about an axis parallel 
to the axis of the rivet. If a rivet, such as A, Fig. 7, be 
considered, the metal on that side of the hole nearest to 
the line BC will be in the condition approximately of a beam 
fixed at each end of the diameter of the hole parallel to BC, 
the bearing load jdt being the load resting upon it and 
assumed to be uniformly distributed over the span d. 
Manifestly the depth of this beam is not uniform, but it 

is assumed to have a depth h — , Fig. 7, throughout the 

2 

span d. If Hs the thickness of the plate, p the pitch of the 
rivets and T the mean intensity of tension between the 
rivet holes, the load on this beam will be {p—d)Tt and the 
moment of inertia of the cross-section will be 



/(- 'J^' 



,. >--^ 



12 

It will be shown in the chapter on bending that k may 

here be taken at -T. 
2 



Art. 72.] DISTRIBUTION OF STRESS IN RIVETED JOINTS. 443 

In Art. 30 the moments at the centre and end of a span 
fixed at each end and uniformly loaded were shown to be 
yV of the load into the span for the end moments and -^V 
of the load into the span for the centre moment. 

Hence, by the usual formulae, 



<-f)' 



M=—{p-d)Tt = -^^^=^T 
(h-fj ' 

■ /. h=-o.sSV(p-d)d+o,sd . ... (5) 

Shearing of Rivets. 

The shearing of the rivets in a riveted joint takes place 
in the plane of the surface of contact between any two 
plates tending to move in opposite directions. In Fig. 8 
the plane of shear would be the surface of contact between 
the main plates A and B, and in Fig. 7 on both sides of the 
main plate, F, i.e., between the main plates E and F and 
at the surface of contact between the main plate F and the 
bent cover-plate D. It is assumed that the total shear is 
divided uniformly between all the shear sections of the 
rivets so that if n were the total number of rivets carrying 
the load P and if d be the diameter of the rivet while 5 is 
the intensity of shearing stress in the normal sections of 
the rivets, there would result for single shear the expression 
P =n.'jSs4d^S. The rivets shown in Fig. 8 and Figs, i, 2, 
and 3 of the preceding article are in single shear. If each 
rivet must be sheared at two normal sections in order 
that the joint may fail (by shear), as in Figs. 4, 5, and 6 
of the preceding article, the rivets are said to be in double 
shear. In the latter case in the preceding expression 2n 
must be written for n for all rivets in double shear. In 



444 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

Fig. 7 the two lower rows of rivets are in double shear and 
the upper row in single shear. 

In Fig. 8 and in Figs. 5 and 6 of the preceding article, 
each row of rivets is assumed to take half the total load 
carried by the joint. That condition, if the cover-plates of 
Figs. 5 and 6 are of half the thickness of the main plates, 
makes the intensity of stress the same in the main plate 
and in the two covers between the two rows of rivets on 
either side of the joint. If, however, the thickness of the 
cover-plate is greater than one-half the thickness of the main 
plate, as is always the case in such joints, then if each row 
of rivets carries half the load, the intensity of stress in the two 
covers between each two rows of rivets will be less than in 
the main plate causing the rate of stretch in the latter to be 
greater than in the former. This condition would throw 
more than half the load, as shear, on the outer row of rivets. 
In other words, the tendency will be to make the stretch 
of the plates within the joint added to the distortion due 
to bending and shearing of the rivets equal to each other 
between each pair of rows of rivets parallel to the joint 
line between the main plates. If again there are three or 
more rows of rivets on either side of an abutting joint, 
there will be a corresponding tendency to overload the 
outer rows of rivets and relieve those nearest the centre 
or abutting line of the joint. There are further conditions 
in addition to those already discussed, militating against 
perfect uniformity in the stress conditions of the complete 
joint. It is impossible, however, to make allowance for 
these complicated and more or less obscure stress con- 
ditions in the operations of design or development of 
formulae. Hence, as already indicated, the usual assump- 
tions of uniformity in the three principal methods of failure 
of riveted joints are made leaving the working stresses to 
be determined by- the results of tests of actual joints. 



Art. 73.] DIAMETER AND PITCH OF RIVETS. 445 

Art. 73. — Diameter and Pitch of Rivets and Overlap of Plate. 
Distance between Rows of Riveting. 

Diameter of Rivets. 
The "diameter of rivet may at least approximately be 
expressed in terms of the thickness of the plate which it 
pierces. There are various arbitrary or conventional 
rules based upon this method of determining the rivet 
diameter. If the unit is the inch, the diameter d may be 
expressed as ranging between the two following values 
for ordinary thicknesses of plate : 

^ = .75^ +.375.) .. 

in which / is the thickness of the plate. Unwin gives the 
following expression for the diameter of somewhat different 
form from that which precedes: 

d = i.2Vt. . . . . . . '(2) 

Neither of the preceding expressions can be applied 
for all thicknesses of plates. If the thickness is great, 
those expressions make the diameter of the rivet too large, 
the diameter rarely exceeding i inch even for the heaviest 
plates. The application of eq. (i) to different thicknesses 
of plates will give the following diameters of rivets ex- 
pressed by the nearest yV in.: 



t 


d 


iin. 


T'ein 


1 


i 


4 


i 


1 


« 


f 


nV 


i 


li 


I 


lA 



446 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

In structural work for ordinary thicknesses of metal 
the prevailing diameters of rivets are f in. and | in. For 
light work, such as sidewalk railings or light highway 
construction, rivets as small as J in. or f in. in diameter 
are used. On the other hand, i to i|-inch rivets are 
employed for specially heavy sections. 

Pitch of Rivets. 

It is possible to determine the pitch of rivets approxi- 
mately by an equation expressing equality between the 
tensile resistance of the net section between two adjacent 
rivets and the shearing or bearing capacity of a single rivet, 
but it is scarcely practicable to proceed in that manner 
as a rule. Again, the pitch will vary to some extent with 
the number of lines of riA^eting on either side of the joint. 
In single-riveting the pitch must be less than. in double- 
or . other multiple-riveting. In boiler or other similar 
riveting, also, the pitch must be usually less than in struc- 
tural work, as questions of steam- and water-tightness or 
other similar considerations are involved in the former 
class of joints. Finally, the pitch will also obviously 
depend largely upon the thickness of plates. In single- 
riveting for comparatively thin plates the following rela- 
tion may be taken, p being the pitch in inches : 

^ = 1,75 in. to 2.25 in. ..... (3) 

For comparatively thick plates in single -riveting the follow- 
ing relation may hol:I: 

^ = 2.375 in. to 3 in (4) 

In double-riveting, p and t still being the pitch and thick- 
ness respectively, the following relation may be taken for 
comparatively thin plates. 



Art. 73.] DIAMETER AND PITCH OF RIVETS. 447 

^=2.6875 in. to 3.25 in (5) 

Again, for comparatively thick plates in double-riveting, 

P=3-37S in. to 3.75 in (6) 

The values given by eqs. (3) to (6) are for boiler or 
other similar work. 

In structural work the pitch of rivets is seldom less than 
about three times the diameter of the rivet, and it is fre- 
quently specified not to exceed sixteen times the thickness 
of the thinnest plate pierced by the rivet. 

Overlap of Plate. 

The overlap of a plate, h in Fig. 2, Art. 71, in a riveted 
ioint is the distance from the edge of the plate to the centre 
line of the nearest row of rivets. This distance, like other 
elements of riveted joints, will depend somewhat upon 
the thickness of the plate as well as the diameter of rivet 
and other similar considerations. It is a common practice 
to make the overlap not less than about 1.5^^, d being the 
diameter of the rivet. Occasionally in riveted joints it 
is made a little less, but 1.5 times the diameter of the rivet 
is about as small as the overlap should be made. Some- 
times J in. is added to the preceding value of the overlap. 

The width of overlap (h) may also be determined in 
terms of d by the aid ofeq. (11) of Art. 72. Since the load 
on the rivet is represented by (p—d)Tt, p must be taken 
in terms of d for a single-riveted joint, in which p = 2^d to 
2|(i. As a margin of safety, and as it will at the same 
tim.e .c-'mplify the resulting expression, let p=sd. 

Eq. (5) of Art. 72 then gives, in confirmation of the 
preceding rule, h = i.sid^ (7) 

* In corsequence of the direct tension in the metal on either side of the 
rivet this value of h should be increased, i.e., to perhaps 1.51^. 



448 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

Experience has shown that this rule gives ample strength, 
and is about right for calking in boiler joints. 

It is to be remembered that the preceding conventional 
rules for the diameter of rivet, pitch, and overlap of plate 
are necessarily to a large extent conventional or approxi- 
mate, and in special cases they cannot be applied with 
mathematical exactness. As practical rules, how^ever, 
they are sufficiently close to give good general ideas of 
those features of riveted joints. 

Distance between Rows of Riveting. 

The distance between the rows of riveting is not susceptible 
of accurate expression by formulae, although the considera- 
tions involved in the establishment of eq. (ii) of Art. 72 
would lead to an approximate value. It is evident, how- 
ever, that this distance should never be as small as h. 
Apparently, in more than double-riveted joints, this dis- 
tance should increase as the centre line of the joint is 
receded from, in consequence of the bending action of the 
rivet. There are other reasons, ho^vever, besides that of 
inconvenience, why such a practice is not advisable. 

In chain riveting the distance between the centre lines of 
the rows of rivets may be taken equal to the pitch in a single- 
riveted joint, or, as a mean, a/ 2.5 the diameter of a rivet. 

In zigzag riveting (Fig. 5) this distance may be taken at 
three quarters its value for chain riveting. 

Art. 74. — Lap-joints, and Butt-joints with Single Butt-strap for 

Steel Plates. 

A butt-joint with single butt-strap, similar to that shown 
in Fig. 3, Art. 71, is really composed of two lap-joints in 
contact, since each half of the butt-strap or cover-plate 



Art. 74.] LAP-JOINTS AND BUTT-JOINTS. 449 

with its underlying main, plate forms a lap-joint. It is 
unnecessary, therefore, to give it separate treatment. 

From these considerations it is clear that the thickness 
of the butt-strap or cover-plate should be at least equal to 
that of the main plate ; it is usually a little greater. 
' Let t = thickness of plates ; 
d = diameter of rivets ; 
p= pitch of rivets (i.e., distance between centres 

in the same row) ; 
r=mean intensity of tension in net section of. plates 

between rivets; 
T' = mean intensity of tension in main plates ; 
/=mean intensity of pressure on diametral plane 

of rivet; 
5= mean intensity of shear in rivets; 
n = number of rivets in one main plate ; 
q = number of rows in one main plate ; 
h =lap as shown in Fig. 2, Art. 71. 
If all the dimensions are in inches, then T, T' , /, and 5 
are in pounds per square inch. 

The starting-point in the design of a joint is the thickness 
t of the plate. The rivet diameter may then be expressed 
in terms of t, and the pitch in terms of the diameter. Such 
rules, like those given in Art. 72, may be useful within 
a certain range of application, but they cannot be depended 
upon in all cases. 

The thickness t of boiler-plate depends upon the internal 
pressure, and is to be determined in accordance with the 
principles laid down in Art. 39, after having made allowance 
for the metal punched out at the holes to find the net 
section. 

In truss work the thickness depends upon the amount 
of stress to be carried, and the same allowance is to be 
made for rivet-holes in finding the net section. 



450 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

The relation existing between T and 7' is shown by the 
following equations: 

tip-d)T^tpT'; ,. J,=^, 

or 

T p-d d 

T-^-'-p W 

In order that the joint may be equally strong in refer- 
ence to all methods of failure, the following series of equali- 
ties must hold: 

-tpT =-tip-d)T = nfdt = o.7Ss4nd'S. 

.'. tpr =t(p-d)T=^qfdt = o.'jSs4qd'S. . . (2) 

It is probably impossible to cause these equalities to 
exist in any actual joint, but none of the intensities T\ T, 
/, or 5 should exceed a safe working value. 

The method of failure by tearing through the gross 
section of the main plate is practically impossible under 
ordinary circumstances, and it is neglected in designing 
riveted joints. This neglect is expressed by dropping 
the first member of eq. (2) and thus reaching eq. (3) : 

t{p-d)T=qfdt^o.7Ss4qd'S (3) 

This equation shows that the usual design of a riveted 
joint must provide against failure in three principal ways: 

1. Tearing through the net section of the plate. 

2. Compression of the metal where the rivets hear against 

the plate. 

3. Shearing of the rivets. 

Although these are the three principal methods of 



Art. 74.] LAP-JOINTS AND BUTT-JOINTS. 451 

failure of riveted joints, whatever may be their type or 
form, the proper design of such joints should be so per- 
formed as to afford provision also against the secondary 
stresses caused by rivet bending, bending of the plates, and 
other indirect influences discusssd in preceding articles. 
This latter end is attained by determining the empirical 
intensities T, f, and 5 of eq. (3) by testing to failure actual 
riveted joints in which those secondary stresses exist. In 
that manner the design against the three principal methods 
of failure, described above, will also afford provision against 
the secondary or indirect stresses of rivet and plate bend- 
ing or other similar conditions. The determination of the 
intensities T, f, and 5 by tests of actual riveted joints will 
be fully shown in the following articles. 

It may be stated here, however, that an approximate 
relation between the ultimate intensities of resistance to 
shear and tension for steel has been used in engineering 
practice in accordance with which 

S = .75T (4) 

It will be found hereafter that / may be taken at least 
1.25 T. If these values be substituted in the third and 
fourth members of eq. (3) in which q = 2, there will result 

(i = 2/(nearly) (5) 

This value of d is too large for thick plates. 

The rivet diameter, therefore, for steel plates may be sai d 
to vary from 2t for thin plates to 1.6^ for thick ones, with 
a maximum diameter of ij to i| inches. The distance 
between the centre lines of the rows of rivets may be taken 
at 2.<,d to T,d for chain riveting and three fourths of that 
distance for zigzag riveting. 

The best designed single -riveted lap-joints give from 
55 to 64 per cent, the strength of the solid plates. 



452 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

Well-designed double - riveted lap-joints should give 
Irom 65 to 75 per cent, the resistance of the solid 
plate. 

Equally well-constructed treble- and quadruple-riveted 
joints should have an efficiency of 70 to 80 per cent, of the 
solid plate. 

It is therefore seen that there is little economy in more 
than double-riveting ordinary joints. 

Art. 75. — Steel Butt-joints with Double Cover-plates. 

Butt-joints with double butt-straps or covers differ in 
two respects, and advantageously, from lap-joints and butt- 
joints with a single cover; i.e., in the former the rivets are 
in double shear and the main plates are subjected to no 
bending. The cover-plates, however, are subjected to 
greater flexure than the plates of a lap-joint, for there is 
no opportunity to decrease the leverage by stretching. As 
the covers form only a small portion of the total miaterial, 
these, with economy, may be made sufficiently thick to 
resist this tendency to failure. 

Let f = thickness of each cover-plate ; and let the re- 
maining notation be the same as in Art. 74. The intensity 
of compression between the walls of the holes in the cover- 
plates and the rivets, and the tension in the former, will 
be ignored on account of the excess in thickness of the two 
cover-plates combined over that of the main plate. This 
excess in thickness is required on account of the bending 
in the covers noticed above. 

The thickness of each cover should be from ^ to I the thick- 
ness of the main plates, or f = .625 to .875/. 

The combined thickness of the covers will thus be from 
1.25 to 1.75 that of the main plates. 



Art. 75] BUTT-JOINTS WITH DOUBLE COVER-PLATES. 453 

The four principal methods of rupture in the main plate 
will then lead to the following equations, corresponding to 
eq. (2), Art. 74: 

-tpT =-t(p-d)T ^nfdt = i.S7oSnd'S. 

.'. tpT =tip-d)T=qfdt = i.S7.oSqd'S. . . (i) 

As in Art. 74, and for the reasons there given, the first 
member of eq. (i) may be "omitted, thus giving 

t{p-d)T=qfdt^i.S7oSqd'S (2) 

Tests of steel butt-joints with double cover-plates as 
well as other tests in bearing and tension in net section of 
plates make it reasonable to take / = i . 2 5 7, with T having 
values from 55,000 to 60,000 pounds per square inch for 
thick plates to perhaps 65,000 to 70,000 pounds per square 
inch for thin plates. 

With this value of /, and g = 2 , the first and second 
members of eq. (2) give for double-riveted butt-joints with 
two covers; 

P=3'Sd ....... (3) 

If the same value of / be preserved, there will result for 
single-riveted butt-joints with two covers 

P = 2.sd (4) 

If, as in the preceding article, there be taken 5 = .757 
and /== 1.257, the second and third members of eq. (2) 
give 

d = i.o6t (5) 

This value of the rivet diameter is too small for thin plates, 
but about right for thick plates. 



454 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

Double-riveted butt-joints designed in accordance with 
the foregoing deductions should give a resistance ranging 
from 65 to 75 per cent, of that of the solid plate. 

Single-riveted joints will give an efficiency somewhat 
less; perhaps from 60 to 65 per cent. 

It is to be supposed, in applying the rules just established, 
that all steel plates are drilled or punched and reamed. 

As in the preceding cases, the distance between the 
centre lines of the rows of rivets may be taken at 2.5 to ^d 
for chain riveting, and three quarters that distance for 
zigzag. 

Art. 76. — Tests of Full-size Riveted Joints. 

There have not been many tests of full-size riveted 
joints of either iron or steel, and those which have been 
made seldom include such heavy steel plates as are now 
frequently employed both in boiler work and for structural 
purposes. The most valuable tests available and with the 
greatest range in size of r vet and thickness of plate are 
those which have been made at the U. S. Arsenal, Water- 
town, :\Iass. The results shown in Table I were taken 
from ''Senate Ex. Doc. No. 1,47th Congress, 2d Session," 
while those in Table II are taken from "Senate Ex. Doc. 
No. 5, 48th Congress, ist Session." The results shown in 
Table III are from the same source and are given in the 
''U. S. Report of Tests of .Aletals and Other Materials" 
for 1896. The character of plates, rivets, and holes is 
shown in the tables, and the intensities of tension in the 
net sections of plates, compression or bearing on diametral 
surface, and shearing on riA^ets are those which existed at 
the instant of failure. The bold-face figures show the 
kind of failure, and when such figures are found, for the 
same test, in two or three columns, they show that the 
same two or three kinds of failure took place simultaneously. 



Art. 76.] TESTS OF FULL-SIZE RIVETED JOINTS. 

Table I. 
RIVETED JOINTS— IRON AND STEEL. 



455 



Nq, 



Size of 
Rivet 
and 
Kind. 



Pitch of 
Rivet. 



Maximum Stresses, 


^j 


Pounds per Square Inch. 


G 










Com- 




is 

2 «H 


Tension 


pression 


Shearing 


on Net 


on Dia- 


on 




Area of 


metral 


Rivets 


Plate (T) 


Sxirface 


(5). 


^ 




if). 




W 



Remarks. 



Single-riveted lap-joints; J4-inch iron plates. 



426 

427 

436 

437 

428 

429 

438 

439 

430 

31 

47 

48 

49 

50 



^' 


' iron 




' " 


%' 


' ' 


-/H 


' 




' " 


M^ 


/ •• 


^' 


' steel 


%' 


' " 


i^' 


/ <« 


w 


' " 


'V'lfl 


" iron 


yifl 





8s 


7i«" 


iron 


86 


^/i«" 


" 


617 


^" 


" 


618 


w 




432 


%" 


iron 


433 


Vh" 


" 


434 




" 


43 S 


%" 


" 


87 


7/Jq" 


steel 


88 


%6" 


" 



1% " 
1% " 

I%6 " 



43,230 


76,140 


34,900 


57.7 


45,520 


82,910 


38,640 


61.4 


38,580 


73,260 


34,870 


52.8 


41,790 


79,360 


38,660 


57. 1 


52,160 


65,420 


33,420 


60.6 


54,930 


68,890 


35,200 


64.0 


49,420 


87,670 


39,640 


65.9 


47,260 


83,940 


40,610 


63.1 


45,890 


78,220 


45,300 


60.3 


49,720 


84,660 


48,420 


65.5 


41,09s 


66,778 


44,204 


53.1 


37,500 


60,886 


42,038 


4».3 



^V\-q" punched holes. 



drilled 



%6" 



punched 



drilled 



Single-riveted lap-joints; J4-iiich steel plates. 



%" 


iron 


^/k' 


" 


%' 


steel 


%' 


" 


^A' 


iron 


•>^' 


" 


%' 


steel 


%' 




%' 


" 


O/f^' 


" 


^H 


" •• 


^(^ 


" " 




'• 


%' 


" 



I%6 " 
I%6 " 



46,340 


82,480 


37,890 


53.2 


46,010 


81,780 


37,860 


52.8 


60,250 


107,260 


49,270 


69.2 


59,240 


105,290 


48,750 


68.0 


40,950 


77,870 


36,350 


48.2 


42,370 


80.200 


36,710 


49-6 


63,190 


120,160 


56,100 


74-3 


61,310 


116,090 


52,460 


71.8 


66,860 


90,000 


41,790 


68.8 


70,000 


94,230 


43,750 


72. c 


62,496 


101,180 


65,220 


69.0 


58,338 


94,800 


60,382 


64.8 


60,184 


114,603 


52,742 


70.6 


57,439 


109,650 


50,645 


67.6 



^%6" punched holes. 



drilled 



Double-riveted lap-joints; 






J4-inch plates. 

}4" drilled 



38,53s 


64,120 


43,110 


60.3 


41,750 


69,710 


41,750 


65.3 


50,592 


42,118 


28,691 


65.8 


49,950 


41,660 


28,660 


65.3 



holes. 



%6" punched 



Double-riveted lap-joints; 14-ii^ch steel plates. 



61,510 


54,640 


25,400 


70.4 


60,300 


53,715 


25,530 


69.4 


65,400 


64,600 


30,430 


74-9 


64,600 


63,430 


30,430 


74.3 


56,944 


94,910 


57,910 


76.3 


59,130 


98,360 


61,130 


79-5 



i^e" punched holes. 



^:: 



Double-welt butt-joints; l^-i^ich iron plates. 



615!^" iron 
6i6l^" " 



53,475 
50,959 



67,321 
64,138 



16,944 
16,719 



62. 2 
59-3 



I^Vlq" punched holes. 



456 RIVETED JOINTS AND PIN CONNECTION. 

Table I. — Continued. 



[Ch. IX. 











Maximum Stresses, 


_^- 












Pounds per Square Inch. 


c 
•5 






Size of 












"^ c 




No. 


Rivet 
and 


Pitch of 
Rivet. 


Tension 


Com- 
pression 


Shearing 




Remarks. 




Kind. 






on Net 

Area of 

Plate ( T) 


on Dia- 
metral 
Surface 

(/")■ 


on 

Rivets 

(S). 


c t 








Single-riveted lap-joints; ^-inch iron plates. 


62 


iM6"iron 


2 ins. 




37,460 


60,340 


38,280 


49 -o 


%■''■ punched holes. 


63 




2 




36,130 


58,150 


35,520 


47.2 


" " " 


64 




2 " 




38,190 


60,730 


37,530 


49-7 


" drilled " 


65 




2 " 




36,210 


57,530 


36,050 


47.1 


" 


66 




1% " 




41,750 


54,130 


34,230 


50.0 


punched holes. 


67 




iH 




41,290 


53,400 


34,150 


49-3 




720 


i" 


2^16 " 




61,700 


52,970 


26,180 


60.4 


^^i«" ';. 


721 


i" 


2V16 " 




58,510 


50,220 


24,830 


57-1 


11 <( 3< 




Single-riveted lap-joints; %-incli steel plates. 


SI 


l%6"iron 


2 ins. 




39,220 


63,210 


39,740 


45-4 


%" punched holes. 


52 


" " 


2 




37,700 


60,760 


38,190 


43.6 


" " " 


53 


['. ^^^®^ 


2 " 




55,215 


89,580 


56,430 


64.1 


" " " 


54 




2 




54,740 


88,660 


55,460 


63.5 


" '■' '* 


55 


" " 


1% " 




63,650 


80,930 


50,650 


66.7 


drilled 


S6 


" " 


i-M " 




63,976 


81,600 


50,900 


67.2 


.1 


238 


H" " 


2 " 




65,460 


89,490 


53,560 


70.9 


me" punched " 


239 


I iron 


2 " 




65,210 


88,990 


53,600 


70.6 


" " " 


718 


2%6 " 




73,894 


79,510 


36,614 


71-4 


i^ie" ;; ;; 


719 


i" " 


25/16 " 




73,970 


80,200 


36,590 


72.0 






Double-riveted lap-joints; %-inch iron plates. 


68 


mQ"iTon 


2 ins. 




48,450 


39,160 


24,760 


63.^ 


%" punched holes. 


69 




2 




50,730 


41,070 


26,150 


66.4 


" " " 


S8 




2 " 




50,220 


40,640 


25,330 


65.7 


" " " 


70 




2 " 




46,255 


41,480 


27,550 


60. 5 


" " " 


71 




2 




46,110 


41,270 


27,010 


60.4 


" " " 


81 




3M '; 




30,920 


58,700 


39,130 


50.4 


!! drilled '' 


82 




3M " 




30,130 


57,340 


38,410 


49.1 






Double-riveted lap-joints; %-inch steel plates. 


57 


iVi6"iron 


2 ins. 




62,800 


■ 50,760 


32,3TO 


73- 2 


34" punched holes. 


59 


" 


2 " 




64,720 


52,450 


32.930 


75.2 


" 


60 


" " 


2 " 




63,210 


56,860 


34,710 


73.2 


" !! 


61 


" " 


2 " 




54,930 


49,530 


^0,8 -,o 


65. 8 


" " 


^J 


" steel 


3^4 " 




44,660 


84,460 


52,750 


64.4 


'.'. drilled 1; 


84 




3^ 




43,650 


83,000 


51,845 


63.0 




Reinforced riveted lap-joints; %-inch iron plates. (See figure next page.) 


244 


%" iron 


j 2 ins. 
'' 4 " 


joint 
welt 


38,870 


59,080 


40,360 


67.6 


iMe" drilled hole, %" welt. 


245 


V4" " 


^;: :: 




43,770 


56,640 


34,460 


74.0 


i%6" 


296 


H" " 


/ " " 




44.840 


57,910 


33,890 


75.7 


.. 1^,, .. 


297>^" " 


]:::: 




. 42.680 


55,350 


31,810 


71.9 


" " " 



Art. 76.] TESTS OF FULL-SIZE RIVETED JOINTS. 

Table I. — Continued. 



457 



No. 



Size of 
Rivet 

and ■ 
Kind. 



Pitch of 
Rivet. 



Maximum Stresses, 




Pounds per Square Inch. 


c 




►— 1 J 






Tension 


Com- 
pression 


Shearing 




on Net 


on Dia- 


on 




Area of 


metral 


Rivets 


Plate ( D 


Surface 


(5). 


^ 




(f). 




a 



Remarks. 



Reinforced riveted joints; %-inch steel plates. (See above figiire.) 

1^6" drilled holes. 



246 


%" steel 


i 2 ins. joint | 
) 4 " welt (■ 


62,050 


67,320 


32,960 


89.0 


247 


Vs," " 


{■:.: :: j 


62,880 


68,135 


33,900 


90. 1 


298 


%" iron 


i:;:: , :; \ 


61,020 


67,300 


34,250 


87.8 


299 


%" ■■ 


i:::; :: \ 


61,710 


68,040 


34,750 


88.9 



240 


H' 


' iron 


241 




" 


292 


" 


" 


29^ 


" 


" 


327 




steel 


328 


" 


" 



Single-riveted lap-joints; }4-inch iron plates. 

^%6" punched holes, 
drilled " 



31,100 


41,500 


34,280 


39-8 


31,395 


41,955 


34,960 


39-7 


32,376 


47,850 


38,020 


42.9 


• 33,180 


48,890 


39,220 


44-3 


39,900 


58,880 


47,020 


52.2 


40,500 


59.S00 


47,830 


54-2 



Single -riveted lap-joints; J^-inch steel plates. 



242 


^" 


iron 


243 


H" 


" 


204 


j^ifi 


' " 


295 


i-Ae 


' " 



38,204 


50,940 


41,100 


38.2 


35.915 


47,890 


38,636 


35-9 


60,210 


56,980 


36,770 


51. 2 


49,590 


47,060 


30,540 


42.2 



i%6" punched holes. 



Double-riveted lap-joints; J^-inch iron plates. 



329 54" iron {2 ins. 
635;%" " I2 " 



6i9l"'-^/i6"iron|2 ins. 

62o|l5/x6" " I2 " 



44,320 
42,920 



59,640 
57,950 



25,280 (57.0 (i%6" punched holes 
24,560 I 55.2 J " 



Double -riveted lap-joints; ^^-inch steel plates. 

29,354 I 19,670 I 5 3 . 8 ( I " punched holes. 



64,602 
I 64,519 I 29,371 

Single-riveted lap-joints; 



730 I iron 
731I1" " 



732 
733I 



;2S^ins. 



34.680 
34,230 



47.5 TO 
46,790 



19,670 53' 
19,644 ] 53-8 J '* 

^-inch iron plates. 

35,460 I 44-9 hVifi" punched holes. 
34,930 I 42.0 I " " " 



Double-riveted lap-joints; ^-inch iron plates. 
[2%ins. I 43.580 I 29,740 I 22,960 I 56. 3 I iMe" punched holes. 

\2% " I 45,850 I 31,310 I 23,670 I 59-3 I " 



Single-riveted lap-joints; 5^-inch steel plates, 
j^lt" steel (234 ins. I 49,6^0 | 56,760 I 43,490 I 5 o . 5 I -i ^ie" Piinched holes. 

55I1" " \2% •* I 52,770 I 60,150 1 46,080 I 53.61 " 

Double -riveted lap-joints; %-inch steel plates. 



36(1" steel \2^^ ms. 

■37I1" " 1 2% " 



69,680 

67.100 i 38,300 I 29,340 I 68.3 



30,780 I 30,470 j 70.9 I i^b" punched holes. 
I 29,340 I 



458 



RIVETED JOINTS AND PIN CONNECTION. 



[Ch. IX. 



Table II. 
RIVETED JOINTS— IRON AND STEEL. 



No 









Maximum Stresses, 


. 


Thick- 
ness of 

Plate 
and 

Kind. 






Pounds per Square Inch. 


;3, 


Diameter 












and 


Pitch of 




Com- 




o § 


Kind of 


Rivet. 


Tension 


pression 


Shearing 




Rivet. 




on Net 


on Dia- 


on 


c S 






Area of 


metral 


Rivets 


.ilPL, 








Plate (D 


Surface 


(5). 





Remarks. 



Single -riveted iron lap-joints. 



%" iron 


iVie" iron 


i%ins. 


39,300 


50,850 


33,710 


47 -o 


" " 


" " 


" " 


41,000 


53,050 


35,170 


49 -o 


W " 


54" 


2 


35,650 


47.350 


37,300 


45.6 


%" " 






35,150 


46,690 


36,780 


44.9 



Single -riveted iron butt-joints. 



punched holes. 



%" iron 


Hio'' iron 


2 ins. 


46,360 


72,390 


25,380 


59-9 


%" punched ho 






" " 


46,875 


73,050 


25,450 


60.5 


)4" " 


H" " 


" " 


46,400 


61,940 


24,630 


59-4 


1%6" " 


" " 






46,140 


61,740 


24,310 


59-2 


" " ' 


Vs" " 


i" " 


2H " 


44,260 


60,330 


23,010 


57-2 


iVie" " 




" " 




42,350 


58,080 


22,310 


54-9 




V4" " 


i^" " 


2.9 " 


42,310 


57,000 


21,870 


52.1 


I%6" " 






" 


41,920 


56,540 


22,140 


51-7 





Single-riveted steel lap-joints. 



%" steel 


%" iron 


1% ins. 


61,270 


65,760 


40,390 


59-5 




" " 


" " 


60,830 


65,320 


39,900 


59-1 


y^" " 


me'' iron 


2 


47,530 


44,590 


29,390 


40.2 


%" " 


" " 




49,840 


46,960 


31,070 


42.3 



m^" punched holes. 



Single-riveted steel butt-joints. 



%" steel 


^Vie" iron 


2 ins. 


62,770 


97,940 


31,240 


71.7 


W punched ho 


" " 


1%6" " 


" " 


61,210 


95,210 


31,020 


69.8 


i" 


^" " 


" steel 




68,920 


62,220 


20,370 


57-1 


" " ' 


2" " 


" " 




66,710 


59,580 


19,890 


55.0 


" " ' 


%" " 


i" 


23^ " 


62,180 


71,450 


27,750 


63.4 


1^6" " 




" " 




62,590 


71,930 


27,940 


63-8 




H" " 


i^" " 


2V2 •; 


54,650 


55.610 


23,190 


54 -o 


I%6" " 


" 


" 




54,200 


55,840 


22,810 


53.4 





It is important to notice that in general the highest uhi- 
mate resistances of tension and compression or bearing are 
found with the thin plates, and that those quantities 
diminish appreciably as the thickness of plate increases, 
both for iron and steel. This law is not so well defined in 
reference to the diameter of rivet, if indeed these tests show 
it at all, except for steel. 



Art. 76.] TESTS OF FULL-SIZE RIVETED JOINTS. 459 

The length of these test joints varied from 9.75 to 13 
inches for Tables I and II, and from 10 to 27 inches for 
Table III. 

Although the results of these tables are somewhat 
irregular, they confirm the general accuracy of the relations 
established between the values of T, f, and 5 in the pre- 
ceding articles, as well as other general rules and conclu- 
sions for boiler work. 

Some efficiencies are lower than those given for similar 
joints in Art. 94, but such instances can, by the aid of the 
tables, be traced either to indifferent design or a phenome- 
nally low value of some one of the three resistances. In 
general the results compare well with those given in that 
article. 

The pitches of rivets are seen to be adapted to boiler 
work, being much less than are ordinarily used in bridge 
work; yet the corresponding resistances show what may 
legitimately be done and expected when unusual condi- 
tions demand a departure from ordinary rules. 

Before deducing working intensities for bridge con- 
struction from the preceding results it is to be first ex- 
plained that those results are as given in the government 
reports, and that the net section used is the gross section 
of the plate, less the actual metal removed by the punch or 
drill, with no allowance for deterioration by the former in the 
immediate vicinity of the hole. Again, in Tables I and II 
the diametral bearing surface and the shearing area of the 
rivet are taken to be those of the drill, or a mean between 
the punch and die in case of punched holes. In bridge 
work, in determining the net section, metal is deducted 
for a diameter equal to that of the cold rivet before driving 
plus one eighth of an inch ; and the shearing and bearing 
are computed for the section and diameter of the cold rivet 
before driving. 



46o RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

Table III. 
TESTS OF STEEL-RIVETED JOINTS; ^-INCH PLATES. 







Maximum Stresses: Pounds per 
Squa,re Inch. 


Efficiency 




Joint. 


Rivet. 








of Joint, Remarks. 






Tension on 


Compres- 


Shearing 


Per Cent. 








Net Area 


sion on 


on 










of Plate 


Diametral 


Rivets 










(T). 


Surface (t). 


(S). 






A 


i" steel 


38,940 


57,960 


41,760 


47 • I it" drilled holes. 


B 








39,450 


81,530 


35,560 


57 




C 








62,200 
56,410 
63,000 
59,33.0 


59.950 
77,900 
88,510 
78,900 


22,480 
29,640 
20,930 
29,410 


83.5 
80.3 

!5-5 :: : 

85.3 












55,050 


71,890 


29.850 


79-4 




H 








51,340 
52,150 
62,390 
58,550 


76,550 

50,170 
54,660 
51,350 


36,030 
20,790 

21,530 
20,620 


78 
78.6 

i::] ."■ ■ 








55,030 


67,490 


27,030 


8s. s 





* Joint not fractured. 

A. Double-riveted lap-joint; -^-inch plate. 

B. Double-riveted butt-joint, two splice-plates; ^-in. plate. 

C. Treble-riveted 

H. Quadruple-riveted butt-joint, two splice-plates; g-in. plate. 

The pitch of the outside rows of rivets in joints B, C, and H was double that of the 
adjoining rows. In the same joints one splice-plate was narrower than the other, so that 
it took one less row of rivets on either side of the joint than the other. 

With these explanations in view, the preceding tests 
justify the following working stresses for the plate-girder 
floor-beams and stringers of railway bridges with machine- 
driven rivets. 



Rivet shearing. 



Rivet 



( 10, 
hearing,. | ^g' 



7,500 'bs. per sq. in. for iron. 



000 



14,000 lbs. per sq. in. for iron. 



000 



( 8,000 lbs. per sq. in. for iron. 
Tension in net section of plate i^ ^^^ ,. u n a a ^^^^-^ 

The bearing resistances are taken rather low, especially 
for steel, for the reason that thick plates are frequently 
used in bridge construction, and the ultimate bearing 



Art. 76. TESTS OF FULL-SIZE RIVETED JOINTS. 461 

resistance for them is appreciably less than for the thin 
plates used in most of the preceding tests. 

The preceding working stresses aie based on steel for 
rivets giving from 56,000 to 64,000 pounds per square inch 
tensile resistance, while the steel for plates, in test speci- 
mens, should offer from 58,000 to 66,000 pounds per square 
inch ultimate tensile resistance. 

In the government report from which Table I is ab- 
stracted, can be found a large number of tests made for 
the purpose of determining the proper minimum distance 
from the centres of rivet-holes to the edge of plates. As 
a result of those tests and other experience on the same 
subject, it may be stated that the least distance from the 
centre of a rivet-hole to the edge of a plate may be taken 
at one and one half the diameter of the hole for steel and 
one and five eighths the diameter of the hole for iron, in 
cases where it is important to secure the maximum resist- 
ance of the joint. 

Efficiencies. 

The values of the quantity which has been termed the 
■'efficiency" of the joint, i.e., the ratio of the resistance of a 
given width of joint over that of an equal width of solid 
plate, in the preceding investigations, are those actually 
determined by experiments with the joints themselves. 
They may, therefore, be relied upon. Some values which 
have for many years been considered as standard, but 
which in reality are of a somewhat arbitrary nature, and 
at best belonging to a limited class of joints, have been 
disregarded. 



462 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

The tests of full-size wrought-iron and steel-riveted 
joints exhibited in Art. 76 show, as a rule, that thin plates 
give materially higher efficiencies than thick plates. Al- 
though there are irregularities, single-riveted lap-joints may- 
yield efficiencies running from 50 to 74 per cent, for J-inch 
plates, but dropping to 50 to 54 per cent, for |-inch plates 
and materially lower for 4-inch plates. On the whole, 
the double-riveted lap-joints show somewhat higher effi- 
ciencies than the single-riveted, but not quite the same 
relative differences between J-inch and |-inch plates, the 
values being found more generally between about 60 and 
80 per cent. 

The single-riveted butt-joints of Table II, Art. 76, 
give efficiencies ranging from about 52 to 72 per cent. 

Some unusually high efficiencies are found in Table III 
of the same article for butt-joints, i.e., about 78 to 90 per 
cent. Those high values are due to the special design of 
the joints, and they cannot ordinarily be attained in prac- 
tice, but they show that well-considered designs will yield 
greatly increased efficiencies. 

In general, efficiencies running from 65 to 70 per cent, 
may be considered excellent for the usual conditions of 
practice. 

Art. 77. — Tests of Joints for the American Railway Engineering 
and Maintenance of Way Association and for the Board of 
Consulting Engineers of the Quebec Bridge. 

In ' ' Proceedings of the- American Railway Engineering 
and Maintenance of Way Association," Vol. 6, 1905, there 
are given the results of a series of tests of carbon-steel riveted 
joints and a duplication of that series of tests in both nickel 
and chrome-nickel steel made for the Board of Consulting 
Engineers of the Quebec Bridge by Profs. Arthur N. Talbot 



Art. 7T.] 



TESTS OF JOINTS. 



463 



and Herbert F. Moore of the University of Illinois, also fully 
described in Bulletin No. 49 (191 1) of that institution. 
There were 144 joints tested in the latter two series. 
Furthermore, there were tested in alternate tension and 
compression 16 other nickel-steel joints and the same 
number of chrome-nickel steel joints. 

All the main plates of these joints were 6.5 inches or 
7.5 inches wide with thicknesses from f inch to f inch 
except the 32 joints subjected to compression, for which 
the plates were 2 inches thick. There were 24 lap joints, 
the same number of butt-joints with double covers or butt- 
straps and an equal number each of the same type of 
joint with one filler and two fillers on each side of both 
main plates. The remaining joints for tension loads only 
(7|X|-inch main plates), with the exception of two sets of 
eight each, were also made with one or two fillers, but the 
latter extended beyond the end of the cover far enough to 
take one rivet. 

All rivets were |-inch in diameter, and those driven by 
a hydro-pneumatic riveter were called " shop " rivets while 
those driven by a hand-pneumatic riveter were designated 

Table I. 

CHEMICAL COMPOSITION OF RIVET AND PLATE MATERIAL 





Nickel-steel 
Riveted Joints. 


Chrome-nickel-steel 
Riveted Joints. 


Element. 


Rivet 

Material 
Per Cent. 


Plate 
Material 
Per Cent. 


Rivet 
Material 
Per Cent. 


Plate 
Material 
Per Cent. 


Carbon 


0. 141 

0.0023 

0.037 

0.442 

3-33 


0.258 
0.008 
0.044 
0.700 

3 330 


0.136 
0.038 
0.032 
0.696 
0.986 
0.240 


0. 191 

0.035 
0.042 
0.485 
0.733 
0.170 


Sulphur, 

Phosphorus 


Manganese 


Nickel 


Chromium . . 









464 



RIVETED JOINTS AND PIN CONNECTION. 



[Ch. IX. 



Table II. 

PHYSICAL PROPERTIES OF RIVET AND PLATE MATERIAL 

All stresses in pounds per square inch. 





Nickel-Steel. 


Chrome-Nickel-Steel. 


Item. 


Rivet 
Material. 


Plate 
Material. 


Rivet 
Material. 


Plate 
Material. 


Number of specimens 
tested 


2 


9 
40,200 
51,700 
89,700 

25.0 

55-8 
29,950,000 


2 8 


Elastic limit 




27,200 
36,300 
63,900 

31-7 

59-9 
30,750,000 


Stress at yield point . . . 
Stress at ultimate .... 
Elongation in 2 inches, 

per cent 

Reduction of area, per 

cent .-.■••• 

Modulus of elasticity. 


45.000 
68,500 

33-5 
634 


38,400 
59,000 

35-2 
63.3 









as "field " rivets. The difference in resistance of the shop 
and field rivets was not material. 

Tables I and II show the chemical composition and the 
physical properties of the nickel and chrome-nickel steels 
vised. 

The following statement shows in a condensed form 
the results of the tests. 

Table III. 

Nickel-Steel Joints 

Lbs. per Sq. In. 

Av. Ult. shear shop and field rivets. 52,440 to 60,140 

Max. tension in plates 16,850 to 50,800 

Chrome-Nickel-Steel Joints 

Lbs. per Sq. In. 

Av. Ult. shear in rivets 48,190 to 56,650 

Max tension in plates 16,170 to 49,500 

Carbon Steel (Main, of Way Assoc. Tests) 

Lbs. per Sq. In. 

Av. shear stress 44,940 to 52,060 

Max. tension in plates 15,190 to 48,400 



Art. 77.] TESTS OF JOINTS. 465 

The shearing of the rivets caused the failures of all the 
nickel-steel and chrome-nickel steel joints. 

The "carbon steel " used in the American Railway Engi- 
neering and Maintenance of Way Association tests was low 
basic open hearth material conforming to the specifications 
of that Association. Some of these joints failed by the 
yielding of the plates but the greater part of them failed 
by the shearing of the rivets and the results are all given 
in terms of the maximum shearing stress in the rivets at 
the instant of failure. 

The lower values in the ultimate and final shear 
stresses in these series of tests belong to the longer rivets, 
i.e. to the joints in which fillers were used. This was to 
be expected in consequence of the increased bending in 
those rivets. Indeed, these tests indicate that with ordi- 
nary thicknesses of plates the carrying capacity of the 
rivets begins to be seriously affected when the " grip " of 
the rivets, i.e. the aggregate thickness of plates pierced by 
them, exceeds about four diameters. It should be stated, 
however, that this depends much upon the design of the 
joint. 

Friction of Riveted Joints. 

Careful observations were made by Profs. Talbot and 
Moore as well as in the tests of joints for the American 
Railway Engineering and Maintenance of Way Association 
to determine the friction of riveted joints which experienced 
engineers have long known to exist. These observations 
indicate that a material slipping of the plates took place in 
some of these joints when the shearing stress in the rivets 
was not greater than about 6,000 pounds per square inch. 
In other cases this slipping took place when the rivet shear 
was as high as 15,000 pounds per square inch. It was 
observed, as might be anticipated, that the quality of the 



466 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

material of the joints had little effect upon the degree of 
stress at which slipping began. The results were about 
the same for the low carbon steel joints as for the chrome- 
nickel steel joints. As might be expected in a well-pro- 
portioned joint, the friction between the plates depends 
upon the force with which they are held in contact by the 
rivets. The motion of the plates is obviously due to the 
fact that the shaft of the rivet in cooling contracts more 
than the comparatively cool plate around it leaving a small 
annular space between the rivet and the wall of the hole. 
As the load on the joint increases a degree of direct stress 
of teiision (or of compression in joints under compression) 
is reached at which the plates slip on each other bringing 
the rivet shafts successively, or more or less simultaneously, 
in contact with the bearing side of the hole. 

After the load increases still more, a higher stage of 
stress is reached at which the yield point of the joint is 
found when relatively rapid distortion takes place. As an 
average the yield point of the nickel steel joints was found 
at an intensity of shearing stress in the rivets of about 
35,000 pounds per square inch and not much different from 
that for the chrome-nickel steel joints. Material bending 
of the rivets appears to be an influential element in the 
increased deformation at the yield point of a joint and it 
is reasonable to suppose that, other things being the same, 
the longer the rivets the lower will be the degree of stress 
at which the yield point is found, although it is doubtful 
whether the rivets are long enough in the well-designed 
riveted joints of good engineering practice to show much 
effect upon the yield point. Profs. Talbot and Moore 
state that " The ratio of the yield point of riveted joints 
to ultimate shearing strength in these tests was about the 
same as the ratio of the yield point of the plate material in 
tension to the ultimate tensile strength of the plate material." 



Art. 78.] RIVETED-TRUSS JOINTS. 467 

The results obtained from the joints tested in alternate 
tension and compression were not markedly different from 
those obtained in tension. The yield point seems to be 
slightly lowered after a few alternations of tensile and com- 
pressive loads. If these alternations took place rapidly, 
doubtless the joints would show much diminution of re- 
sisting capacity but the actual alternations were few in 
number and not rapidly made. 

These tests show that the friction between plates of a 
riveted joint cannot properly be considered as enhancing 
the resisting capacity. Furthermore, this slipping has a 
direct bearing upon the computations of secondary stresses 
in trusses with riveted connections. The corresponding 
deformation may militate materially against the accuracy 
or reliability of such computations. 

Art. 78. — Riveted-truss Joints. 

The circumstances ki which riveted joints are used in 
truss work render permissible many special forms which 

^, r\ n\ n) n\ r^ r^ r^ r-\ r\ ^ r-^ r^s 

, 1 . I <o c^. 



i ny-jy I'-tr^n^^cr "h^-^o^o^ 



^^o ojo o^ 
'b^ooiooOo 

Oq 01 O O^/- 



Fig. I. 

can find no place in boiler-riveting. If joints are found 
under the same circumstances, as far as the transference 
of stress is concerned, precisely the same forms would be 
used, except that calking is, of course, only required in 
boiler work. 



-<A«&,. 


C^ D 

D 


C ) 
C ) 



468 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

Fig. I shows a common form of chord construction in 
riveted-truss work, with the relative proportions exaggerated. 

The lower portion of the figure shows 
a section of the chord in which the cover- 
or splice-plate is shaded. The joint is 
supposed to be in tension. 

In this form of joint the splice-plate 
material is reduced to a minimum. These 
are, in reality, two lap-joints CD and DE 
with the two plates C and E to be spliced. 
In each lap-joint there should be sufficient ^ ° 

rivets determined by the methods of Art. 74. The splice- 
plate AB should be long enough to give the requisite plate 
AC to the left of C, with the same length from 5 to a point 
vertically over E. 

In most cases one or two plates only should be spliced 
at the same point. 

The joint in the vertical plate should be formed as at 
EG; i.e., it should be a double-cover butt-joint. The 
principles already established in a preceding section, in 
regard to the thickness of covers and diameter of rivets, 
should be observed here. 

The two or more full rows of rivets on either side of the 
joint may as well be chain-riveted with a pitch of 3 J to 4 
diameters. Other rivets should then be staggered in until 
the group of rivet centres on each side is brought to a point, 
as shown in the upper part of Fig. i. In this manner the 
available section of a width of plate equal to that of the 
cover becomes approximately equal to the total, less the 
material from one rivet-hole. Hence the efficiency of the 
jomt becomes correspondingly increased. 

If the joint is in compression the preceding observa- 
tions hold without change, except that all covers should 
have the same thickness as the plates covered.- 



Art. 78.] RIVETED-TRUSS JOINTS. 469 

Even if the joints C, D, E, and H are of planed edges, 
little or no reliance should be placed upon their bearing on 
each other, since the operation of riveting will draw them 
apart more or less, however well the work may be done. 

Unless great caution is observed and excellence of design 
secured, there will frequently be excessive bending in the 
riveted joints of truss work, on account of the great variety 
of connections required. 

Diagonal Joints. 

Diagonal riveted joints have from time to time been 
proposed, the line of the joint making an angle of perhaps 
45° to the line of action of the loading. Such joints 
when properly designed have high efficiencies for the reason 
that a normal section of the joint taken anywhere within 
its extreme limits will lie wholly within the main plates 
except at the point where the oblique joint line cuts it. In 
designing such a joint, however, the rows of rivets should be 
placed parallel to the joint line and extend across the entire 
main plates, or some other arrangement may be employed 
which will make the centre of gravity of the group of rivets 
on the two sides of the joint lie in the centre line of the 
main plates or other connected members. If this con- 
dition is not attained, there will be eccentricity of the 
aggregate resistance of the rivets on either side of the joint 
line resulting in serious bending about an axis perpendicular 
to the main plates. The added cost of this type of joint 
and the inconvenience of its use in many cases prohibits 
its general employment as a detail in riveted structural 
work. 

Riveted Joints in Angles. 

It has been found by tests of full-size angles that if a 
riveted joint be formed by riveting one leg only, the u]ti- 



470 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

mate tensile resistance per square inch of the net angle 
section may be but 75 per cent, of the ultimate tensile 
resistance of test specimens cut from the same angle. On 
the other hand, if both legs are riveted the ultimate tensile 
resistance per square inch of the net section may easily 
be 90 per cent, of the ultimate resistance of test specimens 
cut from the same angle. These results show that both 
legs of angles should always be riveted at joints. 

Hand and Machine Riveting. 
The development of the pneumatic and other power 
riveters for both shop and field purposes has practically 
eliminated hand riveting from all structural work except 
in rare cases. When hand riveting was done its inferiority 
to power riveting was recognized by specifying that at 
least one-third more rivets should be used when they were 
driven by hand. 

Art. 79. — Welded Joints. 

Welded joints, as a rule, have never been permitted in 
first class structural work. Fairly good joints of that type, 
however, were made where necessary in wrought iron, but 
it is difficult, if not essentially impossible, to make a satisfac- 
tory weld in structural steel by ordinary procedure. In cases 
where welding of steel is done, some method is necessary 
in which the metal at the weld is brought into a state of 
fusion for a material depth. The thermit and other proc- 
esses accomplish satisfactory welded joints in both steel 
castings and rolled bars for many purposes although they 
are not used in structural work. 

Art. 80. — Pin Connections. 

A pin connection consists of two sets of eye-bars or links, 
through the heads at one end of each of which a single pin 



Art. 80. 



PIN CONNECTIONS. 



471 



passes. Fig. i shows a pin connection; A, A, B, B, are 
eye-bars or links, and P is the pin. 




Fig. I. 



The head of the eye-bar (one is shown in elevation in 
Fig. 2) requires the greatest care in its formation. It is 
imperfect unless it be so proportioned that when the eye-bar 
is tested to failure, fracture will be as likely to take place 
in the body of the bar as in the head ; in other words, unless 
its efficiency is tmity. 

In Fig. 2 the head of the eye-bar, or link, is supposed 







'V^^ 




~^\^ 


1 




H ^^^ 


/ 


^.^-^ 


\ 




A 




\ \ 


] 




*/ 




^0 


1 

1 


1 LC 




\ 




1 


Bl 


)] \ 




\ 


^M. 


/ 


/ ; 






/ i 

1 
1 
1— 



Fig. 2. 

to be of the same thickness as that of the body of the bar' 
whose width is w. 

If t is the thickness of the bar so that wt is the area of 
its normal section, then t is generally included between the 



472 RIVETED JOINTS AND PIN CONNECTION. [Ch. IX. 

limits of ^w and |w for ordinary sizes of eye-bars. These 
limits, however, are exceeded both for the smaller sizes used 
and the larger sizes. A bar for which w=3 inches, may 
have a thickness of ij inches, while the maximum thickness 
of a bar i6 inches wide may be no more than 2 inches. 
Similarly the minimum thickness of a 3 -inch wide bar may 
be f inch while the least thickness of a 16-inch wide bar 
ma}^ be taken at i f inches or / = ^w. 

In the early days of eye-bar manufacture earnest efforts 
were made to analyze the complicated condition of stresses 
in the eye-bar head so as to give it a rational outline, and 
an approximate treatment of the problem may be found in 
the "Trans. Am. Soc. of Civ. Engrs." Vol. VI, 1877, the 
results of which agree essentially with those of experi- 
ment. 

After much experimenting, including the thickening of 
the head, it has for many years been the practice to make 
the heads of eye-bars circular in outline as shown in Figs. 
I and 2. In Fig. 2 the front part of the head NBM is a 
semicircle and it is extended on both sides to the left of 
NM so as to be tangent to the circular curves of the neck 
drawn with the radius equal to the width d of the entire 
head. The latter curves are also tangent to the body of the 
bar as shown at H. 

The head is formed by heating the end of the bar to a 
white heat, then upsetting it in a properly-formed die as 
closely as possible to the finished shape. A little finishing 
work is then usually done under a power hammer or between 
rolls. The head is seldom thicker than the body of the bar. 

The normal section of the head taken through the centre 
of the pinhole is usually from 35 per cent, to 40 per cent, 
in excess of the section of the bar. All steel eye-bars are 
thoroughly annealed after the completion of manufacture 
so as to remove all internal stresses in the head and any 



Art. 8o.J PIN CONNECTIONS. 473 

undue hardness that may have been acquired during that 
process. 

The diameter of the pin should never be less than about 
80 per cent, of the width w of the bar, and it may be from 
1 1 to 2 J times that width, the greater of those factors be- 
longing to bars of small width and the smaller to bars of 
the greatest width used. 

In pin connections the pin is subjected to heavy bend- 
ing for which it is carefully designed as well as for the shear 
in its normal section and for the bearing or compression 
between it and the pin hole. The pin and the pin hole 
are accurately machine finished, the diameter of the latter 
being from perhaps ^Jo i^ch (for the smallest pins) to ^ 
inch (for the largest pins) greater than the former. 

If M is the bending moment to which the pin is sub- 
jected, k the greatest intensity of bending stress devel- 
oped, and A the area of the normal section of the pin, eq. 
(4) of Art. 90 gives 

Arl 

M=k^=o.ikd^ {nearly), . . . . (i) 

o 

or 

d = 2.i6JY • (2) 

Values of k, for circular sections, may be found in 
Art. 90. 



CHAPTER X. 

LONG COLUMNS. 

Art. 8i. — Preliminary Matter. 

There is a class of members in structures subjected to 
compressive stress which do not fail entirely by compression. 
The axes of these pieces coincide, as nearly as possible, with 
the line of action of the resultant of the external forces, 
yet their lengths are so great compared with their lateral 
dimensions that they deflect laterally, and failure finally 
takes place by combined compression and bending. Such 
pieces arfe called " long columns," and the application to 
them, of the common theory of flexure, has been made in 
Art. 35. 

Two different formidse w^ere first established for use in 
estimating the resistance of long columns ; they are known 
as *' Gordon's Formula" and ''Hodgkinson's Formula.'' 
Neither Gordon nor Hodgkinson, however, gave the original 
demonstration of either formula. 

The form known as Gordon's formula w^as originallv dem- 
onstrated and established by Thomas' Tredgold ("Strength 
of Cast Iron and other Metals," etc.), for rectangular and 
round columns, while that known as Hodgkinson 's formula 
(demonstrated in Art. 35) was first given by Euler. 

In 1840, however, Eaton Hodgkinson, F.R.S., published 
the results of some most valuable exDcriments made by 

474 



Art. 8 1. J 



PRELIMINAR Y MA TTER. 



475 



himself on cast and wrought-iron columns (Experimental 
Researches on the Strength of Pillars of Cast Iron, and 
other Materials; Phil. Trans, of the Royal Society, Part II, 
1840), and from these experiments he determined empirical 
coefficients applicable to Euler's formula, on which account 
it has since been called Hodgkinson's formula. 

Prof. Lewis Gordon deduced from the same experiments 
some empirical coefficients for Tredgold's formula, since 
which time Gordon's formula has been known. 

The latter has been quite generally used, but it has 
lately been largely displaced by the straight-line formula 
to be given later. Hodgkinson's coefficients and formula 
have now been abandoned. 

Before taking up the subject of long columns it is 
desirable to establish some important properties of the 
moments of inertia of surfaces used in the analytic treat- 
ment of long columns and in some problems of flexure. 

It will also be both convenient and important to de- 
termine the conditions which ex- 
ist with an isotropic character of 
section in respect to the moment 
of inertia. 

In Fig. I let BC be any figure 
whose area is A, and whose cen- 
tre of gravity is at 0. In the 
plane of that figure let any arbi- 
FiG. I. trary system of rectangular co- 

ordinates X', y be chosen and let XY be any other 
system having the same origin; also, let x', y' and %, y be 
the coordinates of the element D of the surface A in the 
two systems. There will then result 




%=%' cos a +y sin a, 
y=^y' cos a — %' sin a. 



476 LONG COLUMNS. [ch. X. 

The moments of inertia of the surface about the axes y and 
X will then be 

j x^dA = cos^ a j x'^dA + 2 sin a cos a J x'y'dA + 

sin^ajys^yl, . . . . (i) 
J y^dA = cos^ aj j'2 j^ — 2 sin a cos a j x'^^'o^yl + 

sin^ a fx'^dA (2) 

If X and y are to be so chosen that they are principal 
axes, then must J xydA =0, or 

o = JxydA = sin a cos aJy^dA + (cos^ a — sin^ a)jx^ydA 

— sin a cos afx'HA ; (3) 

2fx'ydA 



,\ tan 2 a 



fx^'dA-fy'dA 



Hence, since tan 20:= tan (i8o + 2cv), there will always 
be two principal axes 90° apart. 

Now, if j x'ydA =0, while no other condition is imposed. 

tan 20: =0. This makes a=o or 90°; i.e., X'Y^ are the 
principal axes. 

If, however, \ x'y'dA =0, while a is neither o nor .90°, 
eq. (3) becomes 

fy'dA-fx''dA=^o; 

or 

o . . ^ 
tan 2a =-, I.e., mdetermmate. 
o 



Art. 8i. PRELIMINARY MATTER. 477 

This shows that any axis is a principal axis ; also that 
Jx'^dA =fy'dA =fx'HA =Jy''dA. 

Hence the surface is completely isotropic in reference to 
its moment of inertia, or its moment of inertia is the same 
about every axis lying in it and passing through its centre of 
gravity. 

It has been seen that this condition exists where there 
are two different rectangular systems, for which 

fxydA=fx'ydA==Q; 

but the first of these holds true if either :^ or 3; is an axis of 
symmetry, and the latter if either x' or y^ is an axis of sym- 
metiy. 

Hence, if the surface has two axes of symmetry not at right 
angles to each other, its moment of inertia is the same about all 
axes passing through its centre of gravity and lying in it. 

Eqs. (3) and the two preceding it also show that the 
same condition obtains if the moments of inertia about four 
axes at right angles to each other, in pairs, are equal. 

In the case of such a surface, therefore, it will only be 
necessary to compute the moment of inertia about such an 
axis as will make the simplest operation. 

Principal Moments of Inertia. 

If the moments of inertia I' about the axis of Y' and 
I" about the axis of X' be expressed in terms of the prin- 
cipal moments Ii about the axis of Y and 1 2 about the 
axis of X, eqs. (i) and (2) will give by simply changing the 



478 LONG COLUMNS. [Ch. X. 

primes from the second to the first members of the equa- 
tions ; 

Jx"^dA =r =Ii cos 2a+l2 sin2 «. . . . (4) 
^y'HA=r'=l2C0s''cc-\-Iism^a. . . . (5) 

.If the principal moments of inertia Ii and 1 2 are known 
eqs. (4) and (5) show that the moments I' and I" about 
any axes making the angle a with the principal axes may 
at once be computed. 

Adding eqs. (4) and (5) ; 

7^+7''==7i+72=7 (Polarmoment).. . . (6) 

Hence the sum of the two moments of inertia about any 
two axes at right angles to each other is constant and equal 
to the polar moment of inertia. 

If the second members of eqs. (4) and (5) be divided by 
the area A of the cross section, and if the radii of gyration 
be represented by r' , r" , r\ and ^2", 

r^'^ =r2^ cos^ a+ri^ sin^ a (8) 

Each of eqs. (7) and (8) is the equation of an ellipse in 
which r\ is the semi-axis in the direction of the coordinate 
axis X and r^ is the semi-axis of the ellipse in the direction 
of the coordinate axis F, while r' and r ^'are two semi- 
diameters OD' and OD, all as shown in Fig. 2. 

If eqs. (7) and (8) be added, eq. (9) will result; 

/2 4-/^2 ^ri2+r22 (9) 

This equation is the expression of one characteristic of 
the ellipse, viz., the sum of the squares of any two conjugate 



Art. 8i.] 



PRELIMINARY MATTER. 



479 



semi-diameters is equal to the sum of the squares of the 
two semi-axes. The two radii of gyration therefore about 
any two inertia axes at right angles to each other, except 




Fig. 2. 



the principal axes, are semi-conjugate diameters of the 
ellipse. 

Eqs. (7) and (8) are precisely the same in character as 
eq. (3) of Art. 9 and the ellipse of Fig. 2 is constructed pre- 
cisely as was the ellipse of stress. The two principal radii 
of gyration ri and r2 are represented by the semi-axes OA 
and OB, while the semi-conjugate diameters OD^ and OD 
represent the radii of gyration r' and /' taken about any 
two axes at right angles to each other, represented by ON 
and ON'. The construction lines of Fig. 2 show how the 



48o LONG COLUMNS. [Ch. X. 

ellipse is constructed from eqs. (7) and (8), precisely as 
was the ellipse of stress in Art. 9. 

If it is desired to find the radius of gyration about any 
axis, as the semi-diameter OQ, the construction of the 
ellipse shows that it is only necessary to describe the two 
circles with radii ri and r2, as shown in the figure, then 
erect 07V perpendicular to OQ and draw the horizontal and 
vertical lines respectively from N and i^ to their intersection 
D on the ellipse. The semi-diameter OD will be the radius 
of gyration desired and its direction on the figure of the 
cross-section to which it belongs will obviously be ON, i.e., 
at right angles to OQ. 

It is a well-known property of the ellipse that the 
square of the perpendicular p drawn from the center to 
the tangent to the curve, if the inclination of that per- 
pendicular to the semi-axis is q;, is ; 

p^ =ri^ cos^ a-\-r2^ sin^ a. . . . (10) 

This value of p^ is precisely the same as /^ in eq. (7) 
and it shows that the radius of gyration OR = OD about 
any semi-diameter OQ considered as an inertia axis is equal 
to the normal distance between that semi-diameter and the 
parallel tangent RL\ This simple result finds an import- 
ant application in the problem of the fiexure of a beam of 
unsymmetrical cross-section. 

This same normal distance between a semi-diameter 
of the ellipse and the parallel tangent RV is also equal to 

^-^, the semi-major axis of the ellipse being represented 

by Ti and the semi -minor axis by r-z, while / represents 
the semi- diameter. 

The preceding equations indicate the principal proper- 
ties of every form of cross- section which may affect the value 
of the moment of inertia about any axis whatever passing 
through its centre of gravity. 



Art. 82.] 



GORDON'S FORMULA FOR LONG COLUMMS. 



481 




Art. 82. — Gordon's Formula for Long Columns. 

Since flexure takes place in a long column subjected to 
a thrust in the direction of its length, the greatest intensity 
of stress in a normal section of the column may be 
considered as composed of two parts, 
one a uniform compression over the 
whole section the total of which is equal 
to the load on the column, and the 
other the usual uniformly varying stress 
due to flexure the total of which is zero 
and the intensity of which is also zero e[[^ 
along the neutral axis of the section. 
Fig. I, which is supposed to represent ^ 0"^ 

a longitudinal axial section of a column, 
shows completely this composite stress. 
The line fg is the trace of the normal 
section and gd=cf = p' is the uniform 
intensity of compression due to the com- 
pressive load P. The bending moment Fig. i. 
is represented by the stresses of flexure 
varying uniformly in intensity from p" on the right-hand side 
of the section to ef on the left side, being at the neutral 
axis. The compressive stresses are indicated by — and 
the tensile stresses by +. The resultant of these two com- 
posite stresses is a uniformly varying stress with the great- 
est compressive intensity p' -{-p" on one side of the section 
and the small compressive intensity ec on the left side. 
The bending tension neutralizes exactly the same amount 
of uniform compression, making the resultant intensity 
uniformly varying. There is no resultant tensile stress in 
the section, but it is obvious that there would be if the 
bending moment were sufliciently large. In that case fe 
would be larger than Jc. This condition, however, seldom 



482 LONG COLUMNS. [Ch. X. 

occurs in actual structural columns and never unless they 
are slender and too heavily loaded. 

The condition of stress as described above is that ordi- 
narily assumed for columns, but the actual condition of 
stress is frequently, if not almost invariably, much more 
complicated. The details and the different main parts of 
columns do not act with perfect concurrence nor are the 
processes of manufacture even in the most careful shops 
such as to leave the finished members without internal 
stresses, nor are they perfectly straight. In fact the best 
of columns may be a little convex in one direction at one 
part of their length and concave in the same direction at 
another part. It is imperative, however, to have some 
reasonably simple rational analysis on which formulae may 
be based leaving the erratic stress conditions which are too 
obscure and uncertain to be reached by analysis to be 
covered by empirical coefficients determined by tests of 
actual full-size columns and the stress assumptions illus- 
trated in Fig. I fulfill this requisite at least reasonably. 

In order to determine the two parts of the resultant 
stresses show^n in Fig. i, let 5 represent the area of the 
normal section; J, its moment of inertia about a neutral 
axis normal to the plane in which flexure takes place; r, 
its radius of gyration in reference to the same axis; P, the 
magnitude of the imposed thrust ; /, the greatest intensity 
of stress allowable in the column, and J, the deflection 
corresponding to /. Let p' be that part of / caused by the 
direct effect of P, and p" that part due to flexure alone. 
Then, if h is the greatest normal distance of any element 
of the column from the axis about which the moment of 
inertia is taken, by the " common theory of flexure," 



Art. 82.] GORDON'S FORMPLA FOR LONG COLUMNS. 483 

If the column ends are round, ^' = i ; but if the ends 
are fixed, the value of c' will depend upon the degree of 
fixedness. 



Also 



Hence 



, P , „ ^ P/ c'SJh\ 



Eq. (3) may be considered one form of Gordon's formula. 

In order to make eq. (3) workable in actual computa- 
tions, it is necessary to express the deflection J in terms 
of known dimensions of the column. By referring back 
to eq. (6a) in Art. 27 the desired expression for the deflec- 
tion may be found and by its aid, introducing the notation 
of this article, eq. (4) may be at once written; 

It is seen, therefore, that the quantity a^ depends upon 
both p^^ and E., but it is ordinarily considered constant. 
Since I = Sr\ eqs. (i) and (7) give 

cTP PP ^ r n P/ i'\ 

p''=a,^-^a^-,; /./=/ + /'= j(^i +a^j. (5) 

Eq. (8) shows that a^c^ =a. 
Hence 

^ = -^ » (6) 



484 



LONG COLUMNS. 



[Ch. X 




The integration by which eq. (4) is obtained, being 
taken between limits, causes everything to disappear 
which depends upon the condition of the ends 
of the column. Consequently eq. (6) applies 
to all columns, whether the ends are rounded 
or fixed. Let the latter condition be assumed, 
and let it be represented in the adjoining figure. 
Since the column must be bent symmetrically, 
there must be at least two points of contrafiexure. 
Two such points only may be supposed, since 
such a supposition makes the distance between 
any two adjacent points the greatest possible 
and induces the most unfavorable condition 
of bending for the column. 

If B and C are the points of contrafiexure supposed, 
then BC will be equal to a half oi AD, for each half of BC 
must be in the same condition, so far as flexure is concerned, 
as either AB or CD. Also the bending moment at the 
section midway between B and C must be equal to that 
at A or D. Consequently the hinge- or round-end column 
BC must possess the same resistance as the fixed- or fiat- 
end column AD. In eq. (6), therefore, let l = 2BC = 2li, 




Fig. 2. 



P = 



fS 






(7) 



Eq. (7) is, consequently, the formula for free- or round- 
end columns with length h. 

The fiat- or fixed-end column AD is also of the same 
resistance as the column AC, with one end fiat and one 
end round. Hence in eq. (6) let there be put l=iAC=^r, 
and there w411 result, nearly, 



Art. 82.] GORDON'S FORMULA FOR LONG COLUMNS. 485 

P = —^2- ' (8) 

i+i.8a— 

Eq..(8) is, then, the formula for a column with one end 
flat and the other round. A slight element of approxima- 
tion will ordinarily enter eq. (8) on account of the fact that 
C is not found in the tangent at A just as eqs. (6) and (7) 
are based on the supposition that A and D lie exactly in the 
line of action of the imposed load. 

Eqs. (7) and (8) have been and are now generally 
accepted as representing the resistances of columns with the 
end conditions to which they are intended to apply. As a 
matter of fact, however, tests of full-size members have 
demonstrated that those conditions are not realized in the 
actual use of columns. They have further shown that 
essentially but one condition of column ends need be 
recognized, and that is the actual pin-end condition, as 
realized in pin-connected structures. In that condition 
the end of the column is not free to turn . The compression 
between the pin and the metal bearing against it caused 
by the load carried by the column creates a considerable 
surface of contact over a substantial portion of which the 
intensity of pressure is high. This produces a condition of 
great frictional resistance to any motion between the pin and 
the end of the column, but not sufficient probably to induce 
a fixed-end condition. It has been found by test that fiat- 
end columns, as a rule, give less ultimate resisting capacity 
than pin-end columns of the same length and same radius 
of g^^ration of cross-section. This is doubtless due to the 
practical impossibility to secure a central application of 
loading when fiat ends are employed, the resulting eccen- 
tricity reducing the ultimate carrying capacity of the 
members. While, tht^refore, the classes of columns repre- 



486 LONG COLUMNS. [Ch. X. 

sented by eqs. (7) and (8) are still recognized, it would 
be more rational and more in accordance with experience 
to use only the general form of eq. (6) with a determined 
from actual pin-end tests. . 

Although the quantities/ and a, in eqs. (6), (7), and (8) 
are usually considered constant, they are strictly variable. 
Eq. (4) shows that a is a function of p^' -^E. It is by no 
means certain that p" is the same for different forms of 
cross-section, or even for different sections of the same 
form. While the modulus of elasticity E varies slightly it 
may properly be taken as constant. 

Again, the greatest intensity of stress, /, which can 
exist in the column varies not only with different grades of 
material, but there is some reason to believe that it must 
also be considered as varying with the length of the column. 
The law governing this last kind of variation, for many 
sections, still needs empirical determination. It is clear, 
therefore, that both / and a must be considered empirical 
variables. 

Since / and a are to be considered variable quantities, 

p 

let y take the place of / and x that of a; also, let P=-^ 

represent the mean intensity of stress. Eq. (6) then takes 
the form 

P=^. ....... (9) 

in which c = P-^r^. 

In eq. (9) there are two unknown quantities, y and x, 
consequently two equations are required for their deter- 
mination. If two columns of different ultimate resistances 
per unit of section, and with different values of c, are broken 
in a testing machine, and the two sets of data thus estab- 
lished separately inserted in eq. (9), two equations will 



Art. 82.] GORDON'S FORMULA FOR LONG COLUMNS. 487 

result which will be sufficient to give y and x. Those two 
equations may be written as follows : 

y=p\i+c'x), (10) 

y=p-(,+c''x) (11) 

The simple elimination of y gives 

"" c'p'-c"p" ^ ' 

Either eq. (10) or (11) will then give y. 

In selecting experimental results for insertion in eq. 
(12), care should be taken to make the differences p" —p' 
and c' —c" as large numerically as possible, in order that 
the errors of experiment may form the smallest possible 
proportion of the first. 

In consequence of the more or less erratic results of 
tests of full-sized columns, if two or more pairs of values, 
of / and a be found as indicated above, they will not agree 
with each other and some of them may differ largely. Con- 
sequently the procedures illustrated by eqs. (10), (11) and 
(12) are not sufficient for a satisfactory determination of 
the quantities desired. The method of probabilities has 
been employed, but it also is unsatisfactory because of the 
small number of tests available if for no other reason. 

The usual process is to plot the results of tests using - for 

a horizontal coordinate and the mean load per square inch 
of cross-section of column for the vertical ordinate. The 
results of a series of tests will in this manner be represented 
graphically by a more or less extended group of points 

depending upon the range of -. A curved or straight line, 

r 



488 



LONG COLUMNS. 



[Ch. X. 



as the case may be, is then drawn through such a plotted 
group of points so as to give it a mean position among them. 
The quantities / and a are then so determined by trial as 
to produce a curve lying as close as possible to the experi- 
mental curve and the resulting equation will then be 
Gordon's formula for that particular set of tests or type of 
columns. This operation is fully illustrated and will be 
further considered in the next article in connection with 
a series of tests of Phoenix columns and columns of other 
shapes. 

The accompanying diagrams represent some cross-sec- 
tions of columns which have been much used. 



Batten 

-IT'-'I 



I I 



Lattice 



Quebec Bridge 



I IL J 




SQUARE 



J 






TOP CHORD LATTICED. 



NH 



Z-BAR. 



^ir 



SQUARE-LATTICED. 



PLATE AND ANGLES 



There are a large number of other sections which have 
also been employed either for wrought iron or steel columns. 
For large columns it is occasionally necessary to build up 
cross-sections consisting of a number of webs and angles, 
all so secured to each other as to act as a unit. The Quebec 
Bridge section is such a one. 

Occasionally a so-called " swelled " column, i.e. with a 
considerably enlarged cross-section at and in the vicinity 



Art. 82.] GORDON'S FORMULA FOR LONG COLUMNS. 489 

of the centre of the column length, the outline of section 
gradually but not uniformly decreasing from the centre 
towards the ends, is required. A formula for such a 
column similar to Gordon's formula may be written for a 
varying moment of inertia, but it is too complicated to be 
of practical use. In the case of such columns the judgment 
of the engineer must be used in applying a column formula, 
but it will generally be sufficient to take the radius of 
gyration at the middle section of such a member in com- 
puting the racio -. 

The preceding formulae and the considerations on which 
they are based imply without qualification that all parts 
of a column must be so rigidly bound together that each 
such member will act as a perfect unit under loading and 
they include the condition that the cross-section of the 
column is maintained in its proper shape and proportions 
without material distortion up to actual failure of the tested 
columns. It is imperative, therefore, in the design of these 
members that the details, including rivets, lattice bars, 
batten plates and other spacing details, shall be sufficient 
in number and dimensions to maintain the column as a 
unit up to its full carrying capacity. A failure to meet 
these conditions may greatly and perhaps fatally reduce 
the carrying capacity of the column and result in disaster, 
as in the case of the first Quebec Bridge, caused by the 
weak latticing of a compression member. If a column more 
or less weak in its spacing or other details is tested to its 
ultimate resistance, it will yield in some of its weak details 
instead of failing as a whole, i.e., as a unit. 

The general principles which govern the resistance of 
built columns may, then, be summed up as follows. 

The material should be disposed as far as possible from 
the neutral axis of the cross-section, thereby increasing r; 



49© LONG COLUMNS. [Ch. X. 

There should be no initial internal stress; 

The individual parts of the column should he mutually 
supporting; 

The individual parts oj the column should he so firmly 
secured to each other that no relative motion can take place, in 
order that the column may jail as a whole, thus maintaining 
the original value of r. 

These considerations, it is to be borne in mind, affect the 
resistance of the column only; it may be advisable to 
sacrifice some elements of resistance, in order to attain 
accessibility to the interior of the compression member, for 
the purpose of painting. This point may be a very im- 
portant one, and should never be neglected in designing 
compression members. 

Art. 83. — Tests of Wrought Iron Phoenix Columns, Steel Angles 
and Other Steel Colunms. 

During the period of use of wrought iron as a struc- 
tural material many full-size wrought-iron columns were 
tested to failure giving data on which to base long column 
formulae, but as yet few steel columns of full size have 
been tested to failure and the data on which to base proper 
long column formulae, either for ordinary structural carbon 
steel or for nickel steel, are correspondingly meagre. At 
this time (191 5) full-size steel columns are in process of 
testing at the National Bureau of Standards, Washington, 
D. C, and when they are completed, the desired data will 
be much increased. 

In view of this condition of experimental work on steel 
columns it seems best to give the results of tests of an 
extended series of wrought iron Phoenix columns made with 
much care at the U. S. Arsenal at Watertown, Mass. in 
order to illustrate fully the method of graphical treatment 



Art. 83. 



TESTS OF VARIOUS STEEL COLUMNS. 



491 



of such results in the process of seeking proper column 
formulae. The complete account of this series of tests is 
given in the Transactions of the American Society of Civil 
Engineers for 1882 and the numerical data relating both 
to the dimensions of the columns and to the results of the 
tests are given in Table I. It will be noticed that the ratio 



Table I. 



No. 


Length. 


Area. 


r2. 


l-^r. 


/2-r2. 


E.L. 


Exp. 


Pi- 


P'. 


P". 




Feet. 


Sq. In. 


Ins. 






Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


I 


28 


12.062 


8.94 


112 


12,544 




35,150 


32,550 


34,488 




2 


28 


12. 181 


8 


94 


112 


12,544 





34,150 


32,550 


34,488 




3 


25 


12.233 


8 


94 


lOO 


10,000 


27,960 


55,270 


34,000 


35,040 




4 


25 


12. 100 


8 


94 


100 


10,000 




35.040 


34,000 


35,040 




S 


22 


12.371 


8 


94 


88 


7,744 




35,570 


35,420 


35,592 




6 


22 


12. 311 


8 


94 


88 


7,744 





34,360 


35,420 


35,592 




7 


19 


12.023 


8 


94 


76 


5,776 




35,365 


36,800 


36,144 





8 


19 


12.087 


8 


94 


76 


5,776 


29,290 


36,900 


36,800 


36,144 




9 


16 


12 . 000 


8 


94 


64 


4,096 




36,580 


38,130 


36,696 





10 


16 


12 .000 


8 


94 


64 


4,096 





36,580 


38,130 


36,696 




II 


13 


12.185 


8 


94 


52 


2,704 


28,890 


36,857 


39,400 


37,248 




12 


13 


12.069 


8 


94 


52 


2,704 




37,200 


39,400 


37,248 




13 


10 


12.248 


8 


94 


40 


1,600 


26,940 


36,480 


40,700 


37,800 




14 


10 


12.339 


8 


94 


40 


1,600 


28,360 


36,397 


40,700 


37,800 




15 


7 


12 . 265 


8 


94 


28 


784 


29,350 


38,157 


42,200 


38,352 


40,360 


16 


7 


II .962 


8 


94 


28 


784 


29,590 


43,300 


42,200 


38,352 


40,360 


17 


4 


12.081 


8 


94 


16 


256 





49,500 


44,770 





46,300 


18 


4 


12. 119 


8 


94 


16 


256 


28,050 


51,240 


44,770 




46,300 


19 


8 ins. 


II .903 


8 


94 


2.7 


7.29 





57,130 


69,600 




57,140 


20 


Sins. 


11.903 


8 


94 


2.7 


7.29 





57,300 


69,600 




57,140 


21 


25' 2,65" 


18.300 


19 


37 


68.8 


4,733 





36,010 


37,600 


36,666 




22 


8' 9" 


18. 300 


19-37 


24 


576 


29,510 


42,180 


42,840 


■ 


42,160 



of length over radius of gyration ranges from less than 3 up 
to 112 which more than includes the values of that ratio 
in practically all steel structural work. The ends of the 
columns were flat, a condition which usually introduces 
some erratic results, but apparently the care with which 
the columns were tested eliminated this defect. The 
Phoenix column is a particularly advantageous section for 
testing as its different parts are effectively self-supporting 
and furthermore it has a section whose radius of gyration 
is the same in all directions as the latter has two or more 
axes of symmetry not at right angles to each other. 

The numerical quantities in Table I are self-explanatory, 



492 



LONG COLUMNS, 



[Ch. X. 



particularly in connection with eq. (6) of the preceding 
article. 

The five columns in the right-hand half of the Table 
are pounds per square inch for the different purposes shown 
by the headings of the columns, i.e. E. L. represents the 
compressive stress in the column at the elastic lindt, 
while the column headed Exp. indicates the compressive 
load per square inch of section at which it failed in the 
testing machine. The headings, pi, p' and p'' are computed 











1 






































- 


' 
















"n 












1 




i 




























































® 


Wa-te'rtowii Experi 


Tie 


nt 


















































fj 








Mi ' 






























































r 


Bouscarcn 


s 




'^ 




















































d 


1. 








1 1 1 




























































r 






13 


Phoenix 






















































a 




' !' 






































































p . 


y//^\ 






























1 




































^^ 






3 

<£ 
























i 




! 








1 J, 




la 










— i-rT==*^ 


^-1 
























F 




1 


^a 


°^ 


I 

Wat 


ert 


0!o£rii:£otjj^^___ 


^H^ 


=^ 


f^ 


















- 


__ 


S 


^=- 


= 


S 




— 




:~^===^ 




^ 


own Exp, Curve 
































1 


1 










1 






































































1 














































- 




















i 1 1 


1 






















































Flat 


en 


d Ph 


jenix Colum 


IS' 


































































i 




i 
1 




































































1 




_l 


































































1 


! I 










































-J 


















.,L, 


i i 


7-r 1 






















L 











72000 



60000 



48000 



36000 



24000 



12000 



140 



120 



100 



80 



60 



Fig. I. 



values from eqs. (i), (2), and (3) to be explained immedi- 
ately. 

The numerical values in the column headed Exp. are 
accurately plotted in the diagram, Fig. i, by laying off the 

ratios - from to the left as horizontal ordinates and erect- 
r 

ing at their extremities the corresponding ultimate resist- 
ances given in that column as vertical ordinates with the 
scale as shown in Fig. i. It should be observed that 
in the majority of cases in Table I, there are two experi- 
mental results for each value of and each vertical ordinate 

r 

in Fig. I represents the mean of these two results. 



Art. 83.] TESTS OF VARIOUS STEEL COLUMNS. 493 

The full-curved line marked " Watertown Exp. Curve " 
is then drawn so as to represent as accurately as possible 
the actual experimental results which, as shown in the 
figure, include a few tests other than those made at Water- 
town. This experimental curve rises rapidly for small 

values of -, i.e., for what are actually short blocks. At 
r 

the left end of the curve where - equals 140, the slope of the 

curve is but little more than for intermediate values of that 
ratio. 

After. a number of trials it was found that the value of 
pi, SiS given in eq. (i), agrees quite closely with the experi- 
mental curve for all values between - = 28 and - = 112, and 

r r 

the results computed from it are shown in the column headed 
pi of the Table 

/ . 2r' 
400001 I H — - 

^- — V^^ w 



50000 r^ 

Eq. (i) is Gordon's formula for this particular set of 
Phoenix columns except that the value of / (the numerator 
of the second member) is seen to vary slightly with the 

ratio -. In actual engineering practice, however, the 

numerator shown in eq. (i) was displaced by the numerical 
value 42,000, as a constant numerator of the second member 
makes a simpler application of the formula and it was 
siifficiently accurate for all practical purposes. 

Inasmuch as all long columns used in structural work are 

found within the limits of -=30 and - = 120 (usually for 

r r \ J 



494 l^ONG COLUMNS. [Ch. X. 

bridge truss members, loo) Gordon's formula is never used 
outside of practically these limits. 

It may be observed that the experimental curve is 
nearly a straight line from a point just above b to the 
extreme left of the diagram. For that portion of the 
curve, therefore, the following formula applies very closely: 

/?' =39,640 -46-.* (2) 

The results of this formula are given in the column 
headed " ^^" The table, in connection with the diagram, 
shows that this formula may be used with accuracy for 
values of Z-^r lying between 30 and 140, and further ex- 
periments may possibly show that it is applicable above 
the latter limit. 

For values oi l^r less than 30, the following formula 
will be found to give results approximating very closely to 
the experimental curve : 



4 



p =64,700 -4,6ooa/-. ..... (3) 



The results of the application of this formula are given 
in the column headed '' p" .'' 

It will be observed in Table I that the ultimate resist- 
ance per square inch of the Phoenix columns tested for 

* This equation known as the straight-Hne formula for long columns was 
first proposed in a paper by the author before the Annual Convention of the 
American Society of Civil Engineers in 1881. It was established at that 
time concurrently, but independently, by the author and Prof. Mansfield 
Merriman. The formula is sometimes called the Johnson Straight Line 
Formula, but Mr. Johnson's paper, in which he discussed the straight-line 
formula, was not given to the American Society of Civil Engineers until 
1885, four years after the papers by the author and Professor Merriman had 
been published. 



Art. 83.] TESTS OF VARIOUS STEEL COLUMNS. 495 

ratios of - between about 40 and 112 ranges from about 

34,000 to about 38,000 pounds, which is somewhat above 
the yield point of the material but far below the ultimate 
compressive resistance per square inch as found for short 
blocks. 

In built-up sections of columns in which the component 
parts are less well supported than in the Phoenix section, 
the ultimate column resistance per square inch will be but 
little if any above the yield point of the material and with 

high values of - the ultimate resistance may not rise above 
r 

the elastic limit. This is a most important feature of long 

column resistance and it shows the effect of bending or 

flexure which increases as - becomes greater. 

r 

Many tests of full-size pin-end wrought-iron columns 
have shown that, when well designed with lattice bars and 
other spacing details of sufficient capacity, the ultimate 
resistance of such columns may be represented by eqs. (4) 
and (4a) ; 

P= '''Tp (4) 



1 + 



Or; 



30,000 r^ 



^=42,500-140- (4a) 



Although either equation is for columns with pin ends, 
it may be used generally for such end conditions as are 
usually found in structures like bridges or buildings. The 
flat end condition has already been indicated as giving in 
general somewhat erratic results, but with no advantage 



496 LONG COLUMNS. [Ch. X. 

over pin ends for ordinary circumstances or for such ratios 
of - as are commonly employed. 

For working stresses in wrought-iron columns eqs. (5) 
or (5a) may be used. They are derived from eqs. (4) or 
(4a) by dividing the second members of those equations 
by a so-called '' safety factor " of about 3.5; 

11,000 , . 
(5) 



Or; 



30,000 r^ 



^ = 12,000—40-. ..... (5a) 



Steel Columns. 

The paucity of tests of suitably-designed full-size steel 
columns, either with pin ends or other end conditions, has 
already been observed. Some scattered tests of such mem- 
bers have fortunately been made while others have been 
made upon members so designed as to bring out in ex- 
aggerated form certain features of actions of stresses in 
various parts of the columns without, however, reaching 
data available for the best designs for general engineering 
practice. 

Among the most valuable of these data are some results 
of old tests by the late Mr. James Christie and described 
in the Transactions of the American Society of Civil Engi- 
neers for 1884. Mr. Christie tested mild and high steel 

angle struts with ratios of - running from 20 up to 300. 

The mild steel contained from .11 to .15 per cent, carbon, 
while the high steel contained .36 per cent. The ultimate 
tensile resistance of the mild steel ran from 60,000 to 66,000 



Art. 83.] 



TESTS OF VARIOUS STEEL COLUMNS. 



497 



pounds per square inch with 24 to 26 per cent, stretch in 
8 inches. The high steel had an ultimate tensile resistance 
of about 100,000 pounds per square inch and a stretch 
of about 16 per cent, in 8 inches. 

Table II gives the results of these steel angle tests and 



Table IL 
FLAT-END STEEL ANGLE STRUTS. 





Ultimate Resistance, Pounds 




Ultimate Resistance, Pounds 


J_ 


per Square Inch. 


r 


per Square Inch. 


r 


Mild Steel. 


High Steel. 


Mild Steel. 


High Steel. 


20 


72,000 


100,000 


170 


21,000 


26,000 


30 


51,000 


74,000 


180 


19,500 


23,800 


40 


46,000 


65,000 


190 


18,000 


21,800 


50 


43,000 


61,000 


200 


16,500 


20,000 


60 


41,000 


58,000 


210 


15,200 


18,400 


70 


39,000 


56,000 


220 


14,000 


16.900 


80 


38,000 


54,000 


230 


13,000 


15,400 , 


90 


36,500 


51,000 


240 


12,000 


14,000 


100 


35,000 


47,000 


250 


11,100 


12,800 


IIO 


33,500 


43,500 


260 


10,300 


11,800 


120 


31,500 


40,000 


270 


9,600 


11,000 


130 


29,000 


36,500 


280 


9,000 


10,200 


140 


27,000 


33,500 


290 


8,400 


9,500 


150 


25,000 


30,800 


300 


7,900 


9,000 


160 


23,000 


28,300 









Fig. 2 shows the curves formed by plotting in the usual 
manner the ultimate resistances found in Table 11. The 

ratios - are laid off as horizontal ordinates and the corre- 
r 

sponding ultimate resistances as vertical ordinates. These 

curves are highly interesting as exhibiting the various 

stages of resistance offered by columns in compression as 

the lengths increase from small values of - up to large values 

r 

of that quantity. The ultimate resistances decrease rapidly 



498 



LONG COLUMNS. 



[Ch. X. 



-J T T 1 1 1 1 M 1 1 


» JI _ 1 X 1 ni 1 ~r 


15± 1 nn-rr -r 


a. JI 1 Ml J_ 


- ' -t-^ ^H- s 


„ --r- H^ ^T^i ± ^ 


*- 




~| 


\~ 1 


i_ ' / 


1 M ' / _i_ 


1 M / 


1 1 III y 


1 1 1 1 1 / 1 ■ S 


~r 1 "T If n 




1 1 1 M / / 


ill Mill O 


Ml' 1 1 1 1 II / / 


Ml 1 ' ' I li 




-! 1 !-!-!-! l-H-H-l f-^ 


J ^ \ 11 1 1 1! 1 1 / / <=^ 


1 1 II 1 1 i M / / 


1 1 1 M 1 1 / l/t 


1 ~l M 1 1 /I/ 


1 1 1 1 M 1 1 / 


1 1 




MM / 


1 1 i 1 1 1 / 7 


II 1 1 1 M 1 1 1 1 1 1 \ 1 o 


11 1 M 1 II 1 1 // // B 


1 1 11 1 M 1 1 1 1 / / 


1 1 1 ' ^^ ' ' ' ' y' / 




II 1 1 1 1 i 1 M M 1 1 M./l 




1 h 1 1 rij 1 // 1 r 




1 1 1 1 1 1 1 1 1 '/ M y i i o 


1 1 1 i ^ M M // 1 / fe 


' ' ' '• ' 1 //f ' / 




1 ■ M '''/ n 


1 1 1 1 1 2 M //I yf 1 


1 1 1 ' 1 1 '^- ' # ^ -L - 


1 1 1 1 1 1 1 [1 iz/M /' 




i 1 1 J ^M 1 ^' i 




1 M !// |l// 1 -ri 


1 1 1 J/~ 1 // 1 


1 i y\ 1 // 


i jT i "^//f 


1 1 1 1 / ' 1 ' '-•'// 1 1 


1 ! I ' i > i ' 1 '^// 1 


1 n 1 J, /^~' 1 i 1 // 1 M 1 


1 1 1 l^i 1 1 o / 1 1 1 J 




1 1 1 ^'vr n M ^vi 1 1 <= 


1 -^/^ \\ \\ r\ ' h" 


7'f / ' ' ' 




1 ' /I 1 ' ' 1 1 7 M 


1 ^ 1 ' /l 1 1 M ! //"! ll 


1 III // ' M '// 1 M 1 


1 '1 // \ ' // 


'"ill/ 1 i / 


i 1 1 X 1 1/1 


1 1 yif 1 i l>^ 1 1 C5 


1 1/"^ x! 1 1 "^ 


1 >'^ > J 


^ ><^ 1 


■""''^ -^ 1 


^_ ■ — h^ 1 __ -^^^^"^ \ ill 


-^'^ ■ s 1 -'! i i 1 1 1 i i 


^ )-> ! 1 L 1 1 1 i i-s |l 


c/i 2 o 1 S 1 ^J u 


3 o -tj y 3 1 • 1 • 


S _2_ p 1 1 7^" 1 ?^- ! .r 



Art. 83.] 



TESTS OF VARIOUS STEEL COLUMNS. 



499 



when the column ratio - increases from 20 to 40, then up 

to a value of at least 140 the curves differ but little from 
straight lines. Above the latter, the curvature becomes 



ISi^ 



m 



^ 



=? 



m 






I 



«.- 



5 



decided but not sharp and the two lines converge so that 

when- becomes equal to 300 the difference between the two 
r 

resistances is but little over 1000 pounds per square inch. 



500 LONG COLUMNS. [Ch. X. 

This convergence is one element of confirmation of Euler's 
Formula as the carrying capacity for such high values of 

- depends chiefly upon the modulus of elasticity. With 

still higher ratios the two curves would probably coincide 
as both grades of steel have the same modulus. 

The difference between the working parts of the two 
curves show^n in Fig. 2 is reproduced on a much larger scale 

for- in Fig. 3. Between -=30 and -=140, the two full 
r r r 

straight lines may be drawn as shown. As the points 
represent accurately the numerical values of Table II, it 
is seen that the straight lines represent the ultimate resist- 
ances of the angle struts with sufficient closeness for all 

practical purposes between - =35 and - = 140. 

The straight line for the mild-steel angles is represented 
by eq. (6) ; 

^ = 53,000-186- (6) 

r 

Similarly the straight line for the high-steel angles is 
represented by eq. (7); 

^ = 79,000-325- (7) 

The curved broken lines represent approximately the 

unit ultimate resistances for - less than about 40. If the 

r 

second members of eqs. (6) and (7) be divided by a so-called 
" safety factor " of about 3, eqs. (8) and (9) will represent 
working stresses; 

For high steel ^ = 25,000 — 100- (8) 

For mild steel ;/? = 17,000 — 53 - (9) 



Art. 83.] 



TESTS OF VARIOUS STEEL COLUMNS. 



501 



A number of " model " carbon steel columns of large 
dimensions have been tested within two or three years in 
the large testing machine of the Phoenix Bridge Company 
at Phoenixville, Pa., together with two such nickel steel 
columns, under the supervision of Mr. James E. Howard, 
all but three of those tests having been made for the pur- 
pose of affording data for the design of the new Quebec 
Bridge across the St. Lawrence River. The results of these 
tests, as given in the Transactions of the American Society 
of Civil Engineers for 191 1 and in the Engineering Record 
for 1 9 14 are shown on Fig. 3. The average of three tests 
of built up carbon steel columns, 30 inches by 20 inches 



Area, 90.5 Sq. Ins. 
d 



J 



Fig 4. 



42.75 Sq. Ins. 



r 



r 



Fig. 5. 



34.63 Sq. Ins. 



Fig. 6. 



in outline, as indicated by Fig. 4, are shown at d, the value 

of - being 47 and the average ultimate resistance of the 
r 

three tests (varying but little from each other) being 30,000 
pounds per square inch. 

The results shown at ^, / and g are also for carbon 
steel columns with built-up sections shown in the diagram 
on page 488, the cross-sectional area being 70.65 square 
inches. The length of these columns was 18, feet 9 inches 

and the ratio - was 38. 

Again a and b represent results for carbon-steel columns 

having - equal to 78 and 58 and with cross-sectional areas 



42.75 square inches distributed as shown in Fig. 5. 



502 LONG COLUMNS. [Ch. X. 

Finally, the point n represents the result for two nickel 
steel columns having an area of cross-section of 34.63 square 

inches and - = 52, the section being shown in Fig. 6. 

The number of tests of the carbon-steel columns is not 
sufficient to form a proper basis for a straight line long 
column formula, but the broken line drawn through a and 
c and below e may, as a tentative matter, be represented 
by eq. (10); 

^=44,000 — 150- (10) 

All these built-up carbon-steel columns were of mild 
steel, but their ultimate resistances are distinctly lower than 
the results for Mr. Christie's mild-steel angles. Full-size 
tests, however, have shown that the built-up column, unless 
designed with great care so as to act solidly as a unit, will 
not offer ultimate resistances as high as might be expected 
from the quality of the steel of which they are composed. 

On the same basis used for eqs. (7) and (9), the tentative 
working stress for built-up mild carbon-steel columns would 
be ; 

^ = 14,000-50-. (11) 

The average for the two nickel-steel columms, shown at 
n, Fig. 3 is about 50,000 pounds per square inch and more 
than one-third greater than the corresponding result shown 
for the mild carbon steel at c. 

In all these column tests the elastic limit or the yield 
point of the member as a whole appears to be the controlling 
feature, i.e., the ultimate resistance is not above the yield 

point of the column and if the ratio - is comparatively large 

it will not be above the limit of elasticity of the column as 
a whole. It must be remembered also that both the elastic 



Art. 83.] TESTS OF VARIOUS STEEL COLUMNS. 503 

limit and the yield point of built-up columns will be 
materially lower than the corresponding points of a single 
piece of the same metal. 

These tests appear to indicate that the ultimateresistances 
of nickel-steel columns exceed those of mild carbon-steel 
columns in about the same proportion that the elastic limit 
of nickel steel exceeds the elastic limit of the carbon-steel. 

Observations in these tests of full-size columns made 
at Phoenixville by Mr. Howard indicate that steel columns 
may be considered to have a true modulus of elasticity of 
about 29,000,000 or perhaps 29,500,000 for intensities of 
loading not greater than ordinarily allowed working stresses, 
i.e., from 8,000 to 12,000 pounds per square inch. While 
there are not sufncient data to determine precisely such 
physical elements of steel column resistance, there seems 
to be a relative motion of the component parts of a built-up 
member under test, which does not permit the existence 
of a true modulus of elasticity when loadings exceed about 
12,000 to 15,000 pounds per square inch. Obviously the 
more nearly a column acts as a perfect unit, the better 
defined will be its elastic properties. 

Much more data derived from experimental work with 
full-size steel columns are imperatively necessary in order 
to reach definite conclusions regarding actions of stresses 
in the various parts of such members as well as for the 
development of such important details as latticing, battens, 
and other riveted details. 

Typical FormulcB Now in Use. 
As a result of the present conditions of experimental 
knowledge of built columns, as well as of those that are 
not built up, there is a great variety of column formulae 
used by engineers, both of the Gordon and straight-line 
type. The straight-line formula, however, is largely dis- 



504 LONG COLUMNS. [Ch. X. 

placing the Gordon formula. The General Specifications 
for Steel Railway Bridges recomniended by the American 
Railway Engineering Association as applied to the design 
of cross-sections of steel columns is ; 

^ = 16,000 — 70- (12) 

The New York Central Lines are using the same formula 
in the design of their bridge work, as are engineering or- 
ganizations of other railway companies. Under the use 
of this formula a greater compressive load than 14,000 
pounds per square inch is not permitted. 

The American Bridge Company Specifications for Steel 
Structures 1 9 1 3 , uses the following formula in its design work ; 

^ = 19,000 — 100- (13) 

A provision for impact is made and 13,000 pounds per 
sq. in. is the maximum allowed under the use of eq. (13). 

A form of Gordon's formula still appearing in engineer- 
ing practice is 

12,500 

36,000 T"^ 

This formula is really an old wrought-iron column 
formula and should not be used without reducing the 36,000 
in the denominator to 30,000. 

The New York Building Law gives for a steel column; 

^ = 15,200-58 - , (15) 

The formula used by the City of Philadelphia for its 
buildings is of the Gordon type as follows : 

^= Y~-p (^^) 

iH o 

1 1,000 r^ 



Art. 83.] TESTS OF VARIOUS STEEL COLUMNS. 505 

Other formulas could be cited but enough is shown to 
indicate the pronounced lack of uniformity in this practice. 

None of the preceding formulae should be used for - 
less than 30 nor more than about 120. 

In every case where a column formula is used, it would 
be much more convenient to employ a diagram with the 
curves accurately drawn to represent the desired formulae. 
The actual results, without computations, could be read 
directly from such long column curves. 

Details of Columns. 

In addition to the data already given in another portion 
of this article, the tests cited in this chapter show that 
the unsupported width of no plate in a compression member 
should exceed 30 to 3 5 times its thickness. These tests have 
usually been made with plates or metal J to | inch in thick- 
ness, and it is altogether probable that the above ratio 
of width over thickness would be increased with greater 
thicknesses. 

In built columns, however, the transverse distance between 
centre lines of rivets sectiring plates to angles or channels, etc., 
should not exceed 35 times the plate thickness. If this width 
is exceeded, longitudinal buckling of the plate takes place, 
and the column ceases to fail as a whole, but yields in detail. 

The same tests show that the thickness of the leg of an 
angle to which latticing is riveted should, not he less than \ of 
the length of that leg or side, if the column is purely and 
wholly a compression member. The above limit may be 
passed somewhat in stiff ties and compression members 
designed to carry transverse loads. 

The panel points of latticing should not he separated by a 
greater distance than 60 times the thickness of the angle leg to 
which the latticing is riveted, if the column is wholly a com- 
pression member. 



5o6 



LONG COLUMNS. 



[Ch. X. 



The rivet pitch should never exceed i6 times the thickness 
of the outside thinnest metal pierced by the rivet, and if the plates 
are very thick it should never nearly equal that value. 

Art. 84. — Complete Design of Pin-end Steel Columns. 

In actual design it is necessary not only to make appli- 
cation of the preceding formulce for ultimate resistance of 
columns, but also to proportion a considerable number of 
details as matters largely of judgment and experience. If 
the column, like the section shown as the latticed channel 
or latticed upper chord in the preceding article, has two 
open sides as in the former or one open side as in the latter 
latticed, i.e., has small bars of iron running diagonally 
across those open sides in order to hold the parts of the 
column in their proper relative positions, those lattice 
bars vary in size with the size of column. While the dimen- 
sions vary somewhat among engineers, the following table, 
which has been largely used, illustrates effectively sizes 
that may properly be employed. 









For 6 



9 
10 
II 
12 
13 
14 
15 
16 



inch rolled or built channels if Xye- 

" " " " " If Xfe 

" " " '' " ..= If Xf, 

" " " " ..\..ifand-2 XI 
" " " • " if " 2 xt 



19-23 

24-29 

30 



^8 

XI 
XI 
XI 
XI 
XI 
XI 
Xt^, 

x^ 



= 


^r 




h 


a 


-^r 


ir- 


.1" 


" 


If . 


•li 


(( 


2 


•li 


(( 


2h ■ 


•i^ 



Art. 84.] COMPLETE DESIGN OF PIN-END STEEL COLUMNS. 507 



These bars or lattices may be used in single system, in 
which case each one should make an angle of about 60° with 
the centre line of the side of the column on which they are 
placed. If they are used in double system each pair of 
bars will intersect at their mid-points, and in this case the 
bars may make angles of 45° with the centre line of the side 
of the column on which they are emplo3^ed. In the case 
of double latticing the intersecting pairs of bars are riveted 
at their intersections. Lattice bars are held at their ends 
by one rivet or by two rivets according to the size of the 
column, as shown in the next table. 

Figs. I,. 2, and 3 illustrate different modes of riveting 
the ends of lattice bars. The size and number of rivets 




00000000 



00000000 



-ca 



3 



\y UJ 



Fig. I. 




Fig. 2. Fig. 3. 

will obviously depend upon the size of the lattice bars 
employed and to some extent upon the manner in which 
their ends are held. 

The following table has been used in actual structural 
practice and exhibits good practice in the design of single 
latticing. It is based on the supposition that the lattice 
bars are fiats. In very large columns or in some exposed 



5o8 



LONG COLUMNS. 



[Ch. X 



situations it is necessary to use steel angles for latticing, 
the ends of which must be secured by rivets proportionate 
in number and diameter to the size of angle. 



Size of Lattice. 


Rivets: Number 
and Size. 


Number of Rivets 
at Lattice Point. 


Limiting Length of 

Lattice Centre to Centre 

of Inner Rivets. 


ilXx'^andf 


....f" 


I 


13 inches 


2Xt7 




f 


I 


16 ' 




2Xt\ 




1 


I 


10 ' 




2X1 




f 


I 


23 ' 




2X1 




f 


I 


16 ' 




2jXf 




1 


I or 2 


20 ' 




2^X1 




1 


I " 2 


15 ' 




2^Xt\ 




.| 


I " 2 


20 ' 




2hXT\ 




15 


I " 2 


17 ' 




2iXi 




■|- 


I " 2 


26 ' 




2iXi 




tI 


I " 2 


24 ' 




2iX^ 






4 


15 ' 




3X1 






I or 2 


18 ' 




3X1 




•1 


I " 2 


16 ' 




3X1 




^■ 


4 


9 ; 




sXi^ 




|- 


I or 2 


25 ' 




3XtV 




It 


I " 2 


22 ' 




3X1^ 




f 


4 


15 ' 




3Xi 




" 


I or 2 


32 ' 




3Xi 




jf 


I " 2 


29 ' 




sxi 


. 2. . 




4 


21 ' 




sxh 


2. . 


■• 


4 


II ' 


*• 


4XtV 


I . . 


f 


I or 2 


28 ' 


* 


4XtV 


2. . 


4 


22 ' 


1 


4XtV 


2.. i 


4 


15 " 



At each end of the open or latticed sides of the column 
are placed batten plates which limit the latticing. The 
width of these batten plates is determined evidently by the 
width of the column, but the lengths vary somewhat under 
different specifications. A good and convenient rule is 
to make the length of a batten plate at least equal to 
its width. The thickness of a batten plate will depend 
upon the size of column; it is seldom made less than | in. 
and usually not more than f in. for large columns. The 
size of rivet will also depend upon the size of columns. 
Rivets less than fin. in diameter are seldom used in railroad 



Art. 84.] COMPLETE DESIGN OF PIN-END STEEL COLUMNS. 509 

work and rarely more than i in., the prevailing diameter 
being f in. 

One of the most important details of a column is the 
jaw or extension of one side at the end. The two jaws 
contain the pin holes through which are transferred to the 
pin the total load carried by the column. These jaws or 
extensions are formed so as to fit in between the parts of 
intersecting members, usually the upper or lower chords 
and eye-bars. It is, therefore, imperative to make them 
as thin as the bearing upon the pins and the carrying 
capacity of the jaws themselves acting as short columns 
will permit. Figs. 4, 5, 6, and 7 exhibit some types of 




Fig. 4. 

these post jaws as they commonly occur. As the figures 
show, they are formed by cutting away the flanges of the 
angles or channels forming parts of the posts and riveting 
on the pin or thickening plates required to strengthen 
the detail. The jaws form short columns whose lengths 
should be taken from the centre of the pin hole to the last 
centre line of rivets in the body of the column back of the 



5 TO 



LONG COLUMNS. 



[Ch. X. 



cut in the angle or in the flange of the channel. This 
length indicated by / is shown in each of the figures. 
There have been but few tests made to determine the 



-e — e — e— a — e — — ©-^e — q — e- 



-X7 KJ C^ 



^— o 



e — G — e^^-i-^-o 



o o -e — 9 ' o (5 — e 



U 



^ 



Fig. 5. 



o o o a 



I 



.%P^.J$_-Jl_-_S--. 



-x- 



-G-^>- 



H 



Fig. 6. 




Fig. 7. 

resisting capacities of this particular detail, but those which 
have been made form the basis of the following formula 
for medium steel columns. Obviously there will usually 



Art. 84.] COMPLETE DESIGN OF PIN-END STEEL COLUMNS. 511 

be at least two jaws at the end of each column. The width 
of the side of the column will be represented by 6, as shown 
in Figs. 4 and 6, and t will represent the total thickness of 
metal whose width is b, also as indicated in the same figures. 
If P represents the total load on one jaw of the post, usually 
one half the total load carried by the post or column, 
the average working intensity of pressure on the section of 
metal bt may be written 

P I , . 

- = 9000-340- (i) 

The thickness t of metal is usually the quantity desired, 
and eq. (i) gives 

i=-^+\ w 

90000 26 

In these equations P should be taken in pounds, with 
b, ty and / in inches. 

Eq. (2) has been used to a considerable extent in the 
design of steel railroad bridges, and it is probably as reason- 
able and safe a value of the thickness t as can be written 
with the experimental data and experience now available. 
It is applicable to steel with ultimate tensile resistance 
running from 60,000 to 68,000 pounds per square inch. 
For higher steel or for highway bridges, or for other struc- 
tures where less margin of safety may be justifiable, the 
value of t may be made correspondingly less than that 
given in eq. (2). 

Prob. I. It is required to design a mild-steel pin-end 
column 45 feet long between centres of pins to carry a load 
of 353,000 pounds. The column formula to be used is 
essentially that given as eq. (11) of Art. 83 : 

^ = 16,000 — 70— (3). 



512 LONG COLUMNS. [Ch. X. 

This equation gives the greatest mean intensity allowed 
<«n the column, so that p multiplied by the area of cross- 
section to be determined must be 
^ equal or nearly equal to 232,000. 

' if-^ The least diameter or width of a 

built column should not exceed about 
D one thirty-fifth of its length, except 
Pi w^here posts or columns are used as 
■-^ lateral members, when the length may 



,. -t. 



■'I 

r» 






Pj^ g reach as much as 40 times the least 

diameter or width of cross-section. 
In this case the column is to be built of two plates and 
four angles, as shown in Fig. 8, and the width of plate 
FG must, therefore, not be less than about 16 inches. A 
width of 18 inches will make a w^ell-proportioned column 
and that dimension will be assumed. The separation of 
the plates is preferably made such that the moment of 
inertia of the section about the axis AB will be a little larger 
than the moment about the axis CD. The pin will pierce 
the two plates so that its axis will be parallel to CD. Under 
these conditions, if the column is designed so as to be strong 
enough with the moment of inertia of section taken about 
CD, it will be still stronger in reference to the axis AB, and 
no further attention need be given to possible failure about 
the latter axis. 

If columns of this type are proportioned in the general 
manner indicated, the radius of gyration of the section 
about the axis CD will be approximately .35 of the width. 
In this case that trial radius will, therefore, equal 6.3 
inches. Hence, inserting the values of ^ = 540 inches and 
r = 6.3 inches in eq. (3), there will result ;/? = 10,000 pounds 
per square inch. The total area of section required, there- 
fore, will be closely 353,000 -mo, 000 =35.3 sq. ins. The 
distribution of this metal between the plates and angles is 



Art. 84.] COMPLETE DESIGN OF PIN-END STEEL COLUMNS. 513 
largely a matter of judgment. Let there be assumed 

Two 1 8" X \" plates = 22 . 5 sq. ins. 

Four 2>^"X 3F'X 1 1 -pound angles =13 " " 

Total =35 • 5 sq. ins. 

This is a tentative composition of section which must be 
tested by eq. (3) to determine whether it is as nearly 
accurate as it should be. In order to do this, the moments 
of inertia of the section, as indicated, must be taken about 
the two axes AB and CD. 

Moment of Inertia about CD: 

Two iS'^Xf" plates =2X|xS'= 607.50 

Four 3i"X 3F'X i i-lb angles about own axis = 14 . 20 

Four 3i'''x 3J"X I i-lb.angles about CL>=4X3.25X (7-99)^= 829.92 

Moment of inertia = 1451 , 62 

Moment of Inertia about AB: 

i8C4)* 
Two i8"X|" plates about own axis 2X — ^^^ = . 74 

Two i8"X|" plates about A5 2X ii.25X(6.o6)2= 758.70 

Four 3j''X3i"X I i-lb. angles about own axis = 14.20 

Four 3y'X3rXii-lb.anglesabout^i5=4X3.25X(7.38)'= 708.38 

Moment of inertia = 1482 .02 

These computations show, first, that the moment of 
inertia about A 5 is a little larger than that about CD, 
which is as it should be. They also show that the radius of 
gyration r is 6.39 inches. The approximate rule gives r = 
6.3 inches. These two values are sufficiently near to accept 
the former. The trial composition of section may, there- 
fore, be considered satisfactory and final. The thickness 
of the side plates, .625 inch, is sufficient to insure no buckling 
in the unsupported width between rivets. Similarly the 
length of leg of the 3i-inch angles is also far within safe or 
proper limits. All features of the cross-section are, there- 
fore, so arranged as to meet all the requirements of suitable 
resistance in detail. 



514 



LONG COLUMNS. 



[Ch. X. 



The details of the ends of the columns where they are 
formed into jaws, as shown by Figs. 9 and 10, still remain 





\ 



)4 inch 
batten plate 



20- 



-i— 



Fig. 9. 



Fig. 10. 



to be designed. The diameter of pin will be taken at 7 
inches, as shown in Fig. 9. The permissible intensities 
of shearing and of the bearing on the walls of rivet and pin 
holes will be taken as follows: 

Shearing on rivets = 9000 pounds per sq. in. 
Bearing on rivets and pins = 16,000 pounds per sq. in. 

The total thickness of metal in the two post jaws will, 

therefore, be 

_ . . , - . 232000 . , 

Thickness 01 metal = — -— ^ = 2.1 inches. 

7 X 16000 

The thickness of metal in each jaw must therefore be 
at least ly^ inches. Inasmuch as the thickness of side 
plates of the column is f inch, the pin plates to be riveted to 
the side plates must be at least iV ^i^ch thick to supply the 



Art. 84.] COMPLETE DESIGN OF PiN-END STEEL COLUMNS. 515 

proper bearing surface for the pin ; but that thickness must 
be decided by the formula for the jaws, eq. (2). In that 
equation, P = 116,000 pounds, while ^ = 18 inches and /, 
from Fig. 9, is 9 inches. Making these substitutions in eq. 

(2). 

/ = i.i3 inches. 

In order to meet the requirements of the post -jaw for- 
mula, therefore, the pin plate must be at least J inch 
thick. It is essential however to make these details 
specially stiff and strong and the thickness will, therefore, 
be taken at t% inch, as shown in Fig. 9. 

The number of rivets required above the pin hole 
w^ould ordinarily be computed for the thickness of plate 
required for bearing on the pin, i.e., with the thickness of 
pin plate of re inch. Assuming that thickness for this 
purpose, the rivets being taken | inch in diameter, the 
bearing value of a single 'rivet will be 

IXtVX 16,000 = 6125 lbs. 

The single shear of one J-inch rivet at 9000 pounds per 
square inch has a value of 5412 pounds which is less than 
the bearing value; the shear will, therefore, decide the 
number of rivets required. The bearing value of the |-inch 
side plate on the pin is 7X1X16,000 = 70,000 pounds. 
Hence the number of rivets required in the pin plate on 
each side of the column will be 

1 16000 — 70000 . . ^ ^ ^ 

= nme rivets (nearly). 

5412 ^^ 

These nine rivets must be found above the pin. That 
number, however, is far too small for the pin plate acting 
as a part of the jaw, and it will be judicious to make the 
total number of rivets above the pin 12, as shown in Fig, 9. 



5i6 LONG COLUMNS. [Ch. X. 

The jaw plates will extend 5 inches beyond the pin, as 
shown. The two batten plates above which the latticing 
begins will each be taken \ inch thick, and they will be 
placed as shown in both Figs. 9 and 10. 

It is assumed that the ends of the column are to fit 
into or between other members of the truss, so as to require 
cutting away the legs of the steel angles, as shown, as this 
is a common requirement. 

The length of a batten plate should not be less than 
its width. In the present instance the width of batten will 
be 19.75 inches; the length will, therefore, be taken as 
20 inches. 

As indicated in the tabular statement at the beginning 
of this article, the lattice bars, fully shown in Fig. 10, will 
be 2^X1 inches, and the latticing will be taken as double, 
although this is not always done for the size of column in 
this particular instance. The lattice bars will be riveted 
at their intersections also as shown in Fig. 10. The length 
of lattice bar between rivets will be about 11 inches, as 
the angle made by each lattice bar with the side of the 
column will be about 45 degrees. A single |-inch rivet, 
therefore, at the end of each bar will be sufficient, as shown 
by the second table of this article. At each panel point 
of latticing a single |-inch rivet will hold the ends of both 
lattice bars. 

The complete bill of material for the entire column will 
be as follows: 

Four 3i"X 3F'X i i-lb. angles, 46.42 ft. long. . 185 . 7 X 1 1 = 2,043 lbs. 

Two i8''XF plates, 46.42 ft. long 93X38.25 = 3,557 " 

Four 27"Xii"XA"plates 9X21= 189 " 

Four 2o''X 20" xy' battens 6|X34= 227 " 

240 lin. ft. of 2^'' Xf" latticing 240X3.19= 766 " 

1060 I" rivets 10 . 6X 54 = 572 " 

Total weight of one column = 7,354 lbs. 



Art. 84.] COMPLETE DESIGN OF PIN-END STEEL COLUMNS. 517 

Prob. 2. Let it be required to design a mild-steel 
column with pin ends, 36 feet long between centres of pins, 
to carry a load of 225,500 pounds. It is supposed that 
the column is a member of a railroad bridge, so that the 
load given includes a full allowance for impact. Gordon's 
formula as formerly employed in the American Bridge Com- 
pany's specification will be used : 

17000 



1 + 



In this formula p is the greatest mean intensity of 
working pressure allowed on the section of the column, / the 
length between centres of pins in inches, 
and r the radius of gyration of the 1 



I 
--J10-* 



column section in inches. As the length f ""^ 

of the column is but 36 ft. =432 inches J 

two rolled 15 -inch channels latticed 7 ^' 

may be taken as the principal parts, as l 

shown in Fig. 11. By turning to the -^—^ 

tables in any steel handbook, it will be 

found that the radius of gyration of a 

1 5 -inch channel about the axis AB varies from about 5.6 

inches to nearly 5.2 inches. The larger of the two values 

will be tentatively employed. Substituting ^ = 432 and 

r = 5.6 in the above formula for p, 

^ = 11,000 pounds per sq. in. 

Hence the total area required is 

225500 



1 1 000 



20.5 sq. ms. 



The table of steel channels in any handbook shows 
that the combined area of two 15 -inch 3 5 -pound channels 
is 20.58 sq. in., and they will be accepted as correct. The 



5i8 



LONG COLUMNS. 



[Ch. X. 



same table gives the radius of gyration r about the axis 
AB, Fig. II, as 5.57 inches, which is essentially equal to the 
trial value 5.6 inches. 

As shown in Prob. i, it is desirable to have the moment 
of inertia of the section about AB,-Fig. 11, a little less than 
that about CD, the former (AB) being parallel to the axis 
of the pin. Let the separation of the channels be made 
10 inches in the clear. By using the values of the table, 
the moments of inertia about the two axes may be written : 

About Axis AB: 
Moment of inertia = 3 20X 2 =640. 

Hence r^ = -=31.02; .*. r = 5.57 ins. 

20.58 

About Axis CD: 
Moment of two channel sections each about axis parallel to 

CD and through centre of gravity 2X 8 . 48 = 16 . 96 

2 X 10, 29X 5^' =689 . 84 



Moment of inertia = 706 . 80 




Fig. 12. Fig. 13. 

These results are all satisfactory and show that no 
revision of the section as given in Fig. 11 is needed. 

The end details and latticing shown in Figs. 12 and 13 



Art. 84.] COMPLETE DESIGN OF PIN-END STEEL COLUMNS. 519 

remain to be considered. The following data will be 
required : 

Thickness of channel web =-43 inch. 

Allowed shearing on rivets and pins = io,cco lbs. per sq. in. 

Allowed bearing on rivets and pins. . = 20, coo lbs. per sq. in. 

Diameter of rivets =| inch. 

Diameter of pin =6 inches. 

Value of one |-inch rivet in single shear =6,013 lbs. 

Bearing of pin on channel web =6X .43 X 20,000 

— 51,600 lbs. 

Bearing to be carried by pin plate = — ~ — • —51,600 = 61,150 lbs. 

Thickness of pin plate = — = si inch 

6X20000 ^ 

Bearing value of one |-inch rivet on ^-inch plate = 

IXiX 20,000= 8,750 lbs. 

Hence one pin plate needs -- — ^ = ten |-inch rivets. 
6013 " 

It is assumed that the ends of the column must be 
formed into the jaws show^n in Figs. 12 and 13. As indi- 
cated in Fig. 12 the mean or effective length of the jaw is 
12 inches. The load carried by one jaw is 112,750 pounds; 
hence the thickness of that jaw is by eq. (2) 

112750 12 

t = s ::; — + — = itV inch (nearly). 

8000X15 27 ^^ ^ ^' 

The thickness of the jaw or pin plate to be riveted to 
the jaw must therefore be itV— •43=Tf inch. In order 
that these plates may be firmly made a solid extension of 
the post or column they should be riveted to the w^ebs of 
the channels with the rivets shown in Fig. 1 2 . The proper 
design of the jaw, therefore, requires a much longer and 
thicker plate and more rivets than the simple consideration 
of the pin and rivet bearing and shearing. 

The width of channel flange is 3.43 inches, hence the 
total width of column over these flanges, as shown in Fig. 



520 LONG COLUMNS. [Ch. X. 

13, is 16 J inches. Each batten plate is therefore taken as 
17 inches by 18 inches. 

The length of each lattice bar of the single, 30-degree 
latticing will be about 16 inches between centres of rivets 
at their ends. Lattice bars 2 J inches by f inch in section 
will, therefore, be used. 

The complete bill of material for one column will then be 

Two 15'' 35-lb. channels 37i ft. long 2X35X37^ = 2,602 lbs. 

Four 1 3" X 30" X if" plates 10X41 -44 = 415 " 

Four i7''Xi8''Xi"plates 6X28.9 = 173 " 

Forty-six 2^'' Xf'X 19" bars 72X 3- 19 = 230 " 

Two hundred and twenty-five l'^ rivets 2^X 54 = 122 '* 

Total weight of one column . = 3,542 lbs. 

Art. 85.— Cast-iron Columns. 

Cast iron was the earliest form in which the metal 
iron was used for columns, and it is natural, therefore, 
that the first long-column formulae for cast iron should have 
been among the earliest for that class of members. The 
first experimenter was Eton Hodgkinson, who published 
the results of his tests on small cast-iron columns, the 
greatest length of which was but 60.5 inches, in the " Philo- 
sophical Transactions of the Royal Society of London for 
1840." He not only recognized the round- and fixed-end 
conditions, but he also made the distinction between long 
columns and short blocks, the length of the latter being 
from 4 to 5 times the diameter or least cross-section dimen- 
sion. If d be the diameter of the colum.n in inches and / 
the length in feet, and in the case of hollow round columns 
if D be the exterior diameter in inches and d the interior 
diameter in the same unit, while P is the total or ultimate 
load in pounds on the column, Hodgkinson established 
the following formulae for long cast-iron columns: 
J3.76 
-^ = 33> 37977:7 5 (^or rounded ends). . . . (i) 



Art. 85-] CAST-IRON COLUMNS. 521 

(i3 55 

P = 98,9 2 2-77^ ; (for fixed ends) (2) 

For hollow cylindrical columns of cast iron 

P = 29,i2o 77^^ ; (for rounded ends) . . (3) 

/^3-55_^3-55 

P = 99,32o 777^ — —; (for fixed ends) . . . (4) 

The working or maximum load allowed in any design 
of cast-iron columns would be found by taking one fifth to 
one eighth of the values given in eqs. (i) to (4) inclusive. 
It will be observed that Hodgkinson's formulae expressed 
in the preceding equations are simply Euler's formulae 
as given in eqs. (6) and (9) of Art. 35. with the introduction 
of an empirical coefficient and with the indices of d and / 
changed to harmonize with the experimental results. 

As Hodgkinson's experiments were made on very 
small columns of different metal from that used in cast- 
iron columns of the present day, his formulae cannot safely 
be used for practical purposes at the present time. 

A correct formula for cast-iron columns must be based 
upon tests of full-size columns cast with the metal ordi- 
narily employed in structural practice. Such tests have 
been made at the U. S. Arsenal at Watertown, Mass., and 
will be found reported in H. R. Ex. Doc. No. 45, 50th Con- 
gress, 2d Session, and in H. R. Ex. Doc. No. 16, 50th 
Congress, ist Session. A valuable series of tests was also 
made at Phoenixville, Pa., at the works of the Phoenix 
Bridge Co., under the auspices of the Department of 
Buildings of New York City in 1896-97. Although the 
entire series, including both the tests at Watertown and 
Phoenixville, do not cover the variety of sectional forms 
and range of ratio of length to diameter that could be 



522 



LONG COLUMNS. 



JCh. X. 





' Mil] 1 


ill 'i 




-•- ------- - +. i-....| 






_j_----_ -.- _ __-_ — . 




1 


j 






1 










j ! ■ : ■ ■ ' 1 




- - - M -h 1 ! ' / ■ " 






i 1 1 ■ 1 1 1 i ■■♦ 4+ '; ■ ^ 


.... , . , _ ^ , ^ , ,|| . ^ 


i '.•'•! / i 


■ ■ ' . i 7 • • ' 11 




tn I 1 ■ ' 1 


H , / I 


OT = / 1 . / i 


i,\ ' i 1 , 1 : . 1 • / 1 • -^ 1 : 1 / , i : , 


.... . ,, uj 1 • ' 1 ' ■■ 


' 1 1 '^ , ' / 


w ' / 


' O ' / . . ^ ; :<-r, 


! S -1 ' . ' ■ ■ ' , ' CO 


-:'-<•. ./•■■, '1 ' 


z z ■ /■ ■ ^ / ■ ' 


■ ■ ' ■ ' ' 1 i 7 uj •■ : : ' 1 . . 1 J ^ 




'IIxQ: I . •■ / • 


1 ^ < , / 




: z w I J. / m\ ! ' 1 


i < -)• /'•'/• 




: ^ 1 / 


o ■ /'■/■■'■'' ^1 


^ r^ / / i 


^Q / /, i 


i i : . , : 1 ; . . . ' ^ Z . , / , ■ 


1 ; 1 ' : ■ , • , UJ < / / + ' ! 


li.iiMji.'^^, ■/, / '■; 


'■'''' o+ , / 1 


! i 1 i 1 1 ' . , / ^ : 1 




/ / 


' / ■ ' ' 


< ^ ' • 1 / t ■ 


/ / I 




/ ■*■ ' 


1 : ' ' / • ■ 1 ' 


/ / ' ' 


/ / „^ ^ , (- 


. U'.-, .3 : o5 


\ J 


/I ■ 1 


/ ■ ' 


// () + 


/ / 


' / ' 


/ -N , 




' / r/ , 


,i'/ ' --it^ 


/ , x' -. 


■^4'. / / ^ ' 


1 ,-^^/ \A / ^ - 


^^-/ '\/ •^'' ( ^ ■ * 


; ^V^A 7 *-/ i 


' / .<§y '/=:;/:. = i 


XV ^ ^l 


// < / N.,. / ■ : ' •-'5 i , ■ > : 


.^-:^ / ''b/ , c:oi ' ' 


->? ' ^/ ,■-:!.. 


1 "> / dn \ ,, 


' / ■ ^/ ' . ■ ^' . ^, . : ^J ^ ,,-^ , ■ ■ 1 . . 


■/ ■ 1 1 - '-^ ^^ -' ^^ 


.'•.■■/' II' ' '■^ 1 


■ / C-, ( 1 ' ( ' i 


■•/■'' i/ . ! i 


/ ■ ' ^, ■ , i 


-'■'■■■/ ^i . , 


1 •/;■■■■ o / ■ -. I M 


1 : 1 '/ // / ^ ,■'■-:, , 


i . h i ■ . II j . '■ ' ' ' 




li 1 M i i 1 ' ' . ■ i ■ ^/ r ■ ■ ' 


1 M ■ 1 ! i 1 ■ . 1 1 1 i / . ; ' . • ■ . i , , . 1 ••!■ , , 


: 1 1 1 h M 1 1 ;'<'','!, ■ ' '', .::,■,'-.,! i 




, 


. 


1 ( 


__ . ^ — _ j , : 


. ^__ ^ -J L;;._^ ^ r ■ -- ■ - 


.-,. ^-. ^.-^ , ^.^-^ : i^ r 




!'."■ . .1 ^ -J 1 




' ' i ■ ~i , ' , 










-.-... \ , _ _ 1 ._. _. _(■■•_ -^ 


1 ■■ ■ - " i .-. C-. ' .- . ■■■ , , 


" : ] - -1 ' ■ ! _' ' :. i- 


1-^- ■- ^ ■ ^ . ■ \ ] '?, ■ - - ._: i_L 




— ■ ' "■ ' , . : . ! I 




1 1 1 M > i 1 1 1 ' < 




J-M-H- _._ _ -1 1 1 i 1 ! 1 i h , :_ , 1 , , , , .,:,,.■■! i 1 l.y 



Art. 85. 



CAST-IRON COLUMNS. 



523 



desired, the results are sufficiently extended to show closely 
what may be considered the proper ultimate values for 
hollow round cast-iron columns of full size. 



Table I. 









Diameter in Inches. 












Length in 










Area of 
Section 


Length 
over 


Ultimate 


No. 










1 Resistance 




Inches. 


Large End. 


Small End. 


in 


Exterior 


in Pounds 














Square 
Inches. 


Diameter 


per Square 








Inch. 






Ext. 


Int. 


Ext. 


Int. 








I 


190.25 


15 


13 






43.98 


12.7 


30,830 


2 




15 


12.75 






49 


03 


12.7 


27,126 


3 




15 


12.75 






49 


03 


12.7 


24,434 


4 




i5i 


12.75 






49 


48 


12.7 


25,182 




5 




15 


12.66 






50 


91 


12.7 


35,435 


6 


" 


15 


12.63 






51 


52 


12.7 


40,411* 


7 


160 


8 


6 






21 


99 


20 


29,604 


8 


160 


8 


5.91 







22 


87 ■ 


20 


28,229 


9 


120 


6.06 


3.78 






17 


64 


20 


• 25,805 




10 


120 


6.09 


3.96 






17 


37 


20 


26,205 


II 


147-75 


8 


6.5 






17 


08 


18.5 


25,973 


12 


150 


9 


7 






25 


14 


16.7 


21,183 


' 13 


162 


12 


10 







34 


55 


13.5 


30,813 


14 


159 -75 


14 


12 






40 


84 


II. 4 


25,400 


15 


169 


5 


4-54 






3 


5 


34 


29,854 


16 


157 


7.17 


4 


83 






21 


8 


22 


25,470 


17 


157 


6.35 


3 


9 






17 


28 


25 


27,210 


18 


156 


5.8 


4 


03 






13 


22 


27 


25,100 


19 


142.6 


7.68 


5 


52 


5-94 


4 3 


17 


49 


26.7 


29,310 


20 


146.8 


8.01 


5 


58 


5 


9 


4 


35 


18 


65 


21.3 


28,520 


21 


150 


6.17 


4 


85 


5 


09 


3 


48 


12 


08 


27 


33,500 


22 


145.5 


6 


4 


35 


4 


74 


2 


73 


12 


81 


37-1 


24,620 


23 


133.6 


6.02 


4 


36 


4 


84 


2 


88 


12 


87. 


24.6 


28,060 


24 


129.3 


6.03 


4 


35 


4 


87 


2 


95 


12 


87 


23.7 


27,350 


25 


127.6 


7.47 


5 


97 


5 


72 


4 


62 


12 


13 


19.3 


46,660 


26 


118. 5 


3.98 


I 


96 


2 


97 




49 


7 


16 


34-1 


23,090 


27 
28 


119 
118 


3.98 


I 


96 


2 


98 


I 


47 


7 


17 


34.3 


22,040 


3-97 


I 


95 


2 


99 


I 


39 


7 


7^ 


34-2 


25,060 


29 


84.6 


4.88 


3.03 


4 


27 


2 


08 


II 


-5 


18.5 


31,190 



* Not broken. 



Table I shows the results of all these tests, while the 
Plate exhibits the same results graphically. The tests 
Nos. I to 10 inclusive were made at Phoenixville in De- 
cember, 1897, and Nos. 11 to 14 inclusive in 1896; the 



524 LONG COLUMNS. [Ch. X. 

former group under the immediate direction of Mr. W. W. 
Ewing, and the latter under the immediate direction 
of Mr. Gus C. Henning. The results shown for tests 15 to 
18 inclusive were taken from H. R. Ex. Doc. No. 45, 50th 
Congress, 2d Session, but those for Nos. 19 to 29 inclusive 
are either taken or digested from H. R. Ex. Doc. No. 16, 
50th Congress, ist Session, being portions of reports of 
tests of metals and other materials at the United States 
Arsenal, Watertown, Mass. 

As Table I shows, the columns Nos. 19 to 29 inclusive 
were slightly conical, although probably not enough so to 
affect appreciably their resistances. The areas of section 
in square inches for these columns were taken at mid- 
distance between their ends. As the area of section varied 
considerably in some columns that operation may be a 
source of a little error in determining the ultimate resist- 
ance per square inch from the result of the tests, but if the 
error exists at all it must be very small. The mid-external 
diameter was also taken for these columns in determining 
the ratio of the length over the diameter shown in the 
Table and in the Plate. 

As will be observed both in the Table and in the Plate, 
the ultimate resistances per square inch determined by 
the tests are quite variable, even for the same ratio of 
length over diameter. Indeed, in a number of cases they 
are quite erratic. In Nos. i to 6, for which the ratio of 
length over diameter was 12.7, the ultimate resistances 
vary from a little over 24,000 lbs. per square inch to over 
40,000 lbs. per square inch with no failure at the latter 
value. Again, the ultimate resistance per square inch 
for No. 25, which shows a ratio of length over diameter of 
less than 20, is nearly 47,000 lbs. per square inch, which is 
excessively high as compared with other ultimate resist- 
ances with the same or less ratio of length over diameter. 



Art. 85.1 CAST-IRON COLUMNS. 525 

These erratic results are not surprising in view of the 
ordinary character of the metal. It should be remembered 
that the failures of these columns are frequently recorded 
with such "remarks" as the following: "Foundry dirt or 
honey-comb between inner and outer surfaces," "bad 
spots," "cinder pockets and blow holes near middle of 
column," "flaws and foundry dirt at point of break." 
In other words, it was no uncommon feature to observe that 
defects, flaws, or blow holes or thin metal had determined 
the place of failure. There is considerable uncertainty in 
platting the results of tests affected by these abnormal con- 
ditions, but a more or less satisfactory law for the generality 
of cases may be determined from a graphical representation 
of the results, as shown on Plate I. On that Plate the 
ultimate resistances in pounds per square inch, as shown 
in Table I, have been platted as vertical ordinates, while 
the ratios of length over diameter given in the same Table 
are represented by the horizontal abscissas, all as clearly 
shown. The full straight line drawn in about a mean 
position among the results of the tests probably represents 
as near as any that can be found a reasonable law of variation 
of ultimate resistance with the ratio of length over diameter. 
It is evident that within the range of these experiments a 
straight line will represent the ultimate resistances fully 
as well as any curve, if not better, although the results for 
the lengths of thirty-four times the diameter begin to 
indicate a little curvature. The formula which represents 
this straight line, i.e., which gives the ultimate resistance 
per square inch, is as follows: 

/ 
^ = 30,500- 160J. ..... (5) 

It is to be borne in mind that these columns were round 
and hollow, and that thev were tested with flat ends in all 



526 LONG COLUMNS. [Ch. X. 

cases. The ordinary formula, based upon Hodgkinson's 
tests, and frequently used in cast-iron column construction, 
is as follows: 



80000 

1 + 



P = TP ^^) 



400 d 



The curve corresponding to this particular form of 
Tredgold's formula is also shown on the Plate. It will be 
seen that at the ratio of length over diameter of 10 to 12 
(not an uncommon ratio) the ultimate, as given by this 
formula, is just about double that shown by actual test. In 
other words, if a safet}^ factor of 5 were required, as is the case 
in some building laws, the actual safety factor would be but 
2 J. The curve represented by eq. (6) is seen to cross the true 
curve at a ratio of length over diameter of about 29. A 
glance at the Plate will show how erroneous and dangerous 
is the use of the usual formula for hollow round cast-iron 
columns; indeed, that formula is grossly wrong, both as to 
the law of variation and the values of ultimate resistance. 

In view of the working resistances, which have been 
used in the design of cast-iron columns, it is no less interest- 
ing than important to compare the ultimate resistances per 
square inch of mild-steel columns, as determined by actual 
tests, with the ultimate resistances of cast-iron columns, 
as shown by the tests under consideration. The broken 
line of short dashes represents the formula 

I 
^ = 52,000-180- • . (7) 

determined by actual tests of mild-steel angles made by 
Mr. James Christie at the Pencoyd Bridge Works, and 
given in Art. 60. This line or formula shows that the 
ultimate resistances per square inch of mild-steel columns 



Art. 85.] CAST-IRON COLUMNS. 527 

are from 40 to 50% greater than the corresponding quanti- 
ties for cast-iron, the same ratio of length over diameter 
being taken in each comparison. 

When the erratic and unreliable character of cast-iron 
columns is considered, it is no material exaggeration to 
state that these tests show that the working resistance 
per square inch may be taken twice as great for mild-steel 
columns as for cast-iron ; indeed, this may be put as a 
reasonably accurate statement. 

The series of tests of cast-iron columns represented in 
the Plate constitute a revelation of a not very assuring 
character in reference to cast-iron columns now standing, 
and which may be loaded approximately up to specification 
amounts. They further show that if cast-iron columns 
■ are designed with anything like a reasonable and real margin 
of safety the amount of metal required dissipates any 
supposed economy over columns of mild steel. 

If the average working stress per square inch is one 
fourth of the ultimate resistance, eq. (5) gives 

I 
^ = 7600 — 40-7 (8) 

If the working stress is to be taken at one fifth the 

ultimate, eq. (5) gives 

/ 
p = 6ioo-s22 (9) 

In these equations p is the average working intensity 
of pressure in pounds per square inch. The length / and the 
exterior diameter d must be taken both in the same unit, 
ordinarily the inch. 

These formulae may be used between the limits of - = 10 

d 

and -7 = 35 or even 40. They may also be applied to hollow 



528 LONG COLUMNS. [Ch. X. 

rectangular columns with reasonably close approximation, 
d being taken as the smaller exterior side of the rectangular 
cross-section. 

Art. 86. — Timber Columns. 

The greater part of available tests of full-size timber 
columns have been made prior to 1900, and their results 
have not been obtained either by the aid of improved appli- 
ances in testing now employed, or in all respects under the 
care given in later testing work to secure accuracy or to 
avoid misinterpretation of the more or less obscure condi- 
tions which attend the testing of full-size timber members. 

The ratio of the length divided by the radius of gyration 
is much less in timber columns than those of iron or steel. 
Furthermore, as sections taken at right angles to the axes 
of timber columns are almost always rectangular, it is per- 
missible to use the ratio of the length over the least side 
rather than the length over the least radius of gyration, 
gaining thereby a little simplicity in the use of column 
formulae. 

Timber columns are subject to the same vicissitudes of 
knots, wind-shakes, season cracks and decay as other timber 
members. Indeed most failures of full-size timber mem- 
bers are induced by some local defect such as a knot, either 
decayed or sound. Unless in a thoroughly protected place, 
timber columns are in a condition of almost constant change 
and in the long run for the worse. 

The degree of seasoning is an element of material effect 
in the resistance of timber columns. The greater the 
amount of moisture in timber, the less will be its capacity 
for compressive resistance, other conditions remaining un- 
changed. As in all other full-size timber tests, the con- 
dition of moisture should be known and stated in connection 
with the results, of timber column tests. It makes little 



Art. 86.1 



TIMBER COLUMNS. 



529 



or no difference whether the moisture is the original sap or 
the result of a damp atmosphere or immersion in water. 

Among the earliest tests were those of Professor Lanza, 
who investigated timber mill columns, mostly of circular 
section and some of them after standing in use in com- 
pleted buildings for various periods from one year to twenty- 
five years. These columns varied in length from about 2 
to 14 feet, the great majority of them being from 11 to 14 
feet. The diameters varied generally from about 5 inches 
to about II inches. A few were square. Neither the shape 
nor the dimensions of cross-sections appeared to affect 
materially the results of tests. The principal results of 
these tests are given in the tabulated statement below : 





Max. 


Mean. 


Min. 




Lbs. per Sq. In. 


Lbs. per Sq. In. 


Lbs. per Sq. In. 


Yellow pine, partially seasoned 


5,450 


4.370 


3,510 


Yellow pine, air seasoned 


4,892 


4,690 


4,488 


Yellow pine, dock seasoned . . . 


5.950 


4,563 


3,477 


White wood, partially seasoned 


3,333 


3,010 


2,687 


White oak, partialh^ seasoned. 


3.786 


3.070 


1,964 


White oak, in mill 6| years . . . 


6,029 


4,170 


2,945 


White oak, in mill 25 years . . . 


6,147 


4,420 


3,266 


White oak, thoroughly seasoned 


4,450 


3.175 


1,865 



The ends of these columns were usually flat, sometimes 
with a so-called " pintle " or, in a few cases, one end round. 
These results show the usual erratic features of full-size 
timber tests, some of which doubtless are due to undiscovered 
weaknesses at some point. Prof. Lanza stated that " The 
immediate location of the fracture was generally determined 
by knots." Some of the columns were tapered and the 
reduction of the section at the ends of such columns usually 
located the failure at those reduced ends. 

The greatest ratio of length to radius of gyration in 
these columns was about 86, but the actual results did not 
show that there was any discoverable relation between the 



53° 



LONG COLUMNS. 



[Ch. X. 



ratio of the length over the radius of gyration and the 
ultimate column resistance. The latter was influenced 
little or none by the length of the columns. 

Tables I and II show the results of the early tests of 
Col. Laidley, Engineer Corps, U. S. A., made many years 
ago and reported in " Ex. Doc. 12, 47th Congress, ist 
Session." They show the large increase in ultimate resist- 
ance per square inch with short lengths. Indeed some of 
the pieces were short blocks. These results indicate the 
care that should be taken in discriminating between the 
ultimate compressive resistances of short timber blocks and 
long columns. The results in Table I for those pieces 
seasoned twenty years are too high, while those for pieces 
Nos. 16, 17, and 18 are low, in consequence of the material 

Table I. 
YEI.LOW PINE. 



No. 


Length, 
Inches. 


Form of Section. 


Section Dimensions, 
Inches. 


Ultimate Resistance 
per Sq. In. 










Lbs. 


I 


20.4 


Circular. 


10. 2 diam. 


6,676^ 




2 


119-95 


Square. 


II Xii 


6,230 


(U 


3 


119.90 


" 


II Xii 


6,552 


4 


20.0 


" 


10. 4X 10.4 


7,936 


rt 


5 


16.0 


" 


8X8 


8,165 


^ 


6 


8.0 


" 


4X4 


7,394 


73 


7 


3.0 


" 


1.5X 1.5 


5,533 


^1 


8 


6.0 


it 


3 X 3 


8,644 


■"S^ 


9 


6.0 


" 


3 X 3 


8,133 


•§0 


10 


3.0 


" 


1.5X 1.5 


8.389 




II 


3-0 


" 


1.5X 1.5 


8,302 


be 


12 


3-0 


" 


1.5X 1.5 


6,355 


bJO 


13 


14.0 


" 


4-6X 4.6 


9,947 


'3 


14 


17.2 


" 


4-6X 4-6 


10,250 


s 


15 


19. 1 


" 


5-3X 5-3 


7,820 _ 


CO 


16 


180.0 


Rectangular. 


16 XI3-65 


3,070 


17 


180.0 


' ' 


16. 2X 7.0 


2,795 


18 


180.0 




17 X 8.75 


3,180 



Nos. 13, 14, and 15 were pine of very slow growth. 
Nos. 16, 17, and 18 were very green and w^et. 



Art. 86.] 



TIMBER COLUMNS. 



531 



Table II. 
SPRUCE THOROUGHLY SEASONED. 



No. 


Length, 
Inches. 


Form of Section. 


Section Dimensions, 
Inches. 


Ultimate Resistance 
per Sq. In. 


, 








Lbs. 


I 


24 


Rectangular. 


5-4X5-4 


4,946 


2 


24 






5 


4X5 


4 


4,811 


3 


36 






5 


4X5 


4 


4,874 


4 


36 






5 


4X5 


4 


4,500 


5 


60 






5 


4X6 


4 


4,451 


6 


60 






5 


4X6 


4 


4,943 


7 


120 






5 


4X5 


4 


3,967 


8 


120 






5 


4X5 


4 


4,908 


9 


60 






5 


4X5 


4 


5,275 


lO 


30 






5 


4X5 


4 


5,372 


II 


15 






5 


4X5 


4 


5,754 


12 


121 .2 


Circ 


ular. 


12 


4 dia 


m. 


4,681 



being green and wet. The tests pieces in Tables I and II 
were generally fine straight -grained timber of better quality 
than ordinarily used in engineering practice. 

This condition accounts largely for Col. Laidley's results, 
being materially higher than Prof. Lanza's for the same 
kind of timber. 



Formula of C. Shaler Smith. 

Although these formulae were deduced from tests made 
many years ago, they have been so extensively used over 
such a long period that they may properly be considered 
among the classics of engineering literature of this kind. 
Hence they are given here, although not now used so 
widely as formerly. 

The tests of full-size sticks on which the formulae are 
based were grouped by Mr. Smith as indicated and the 
corresponding formulae are as given below. 



532 LONG COLUMNS. [Ch. X. 

*' I St. Green, half -seasoned sticks answering to the 
specification 'good, merchantable lumber.' 

*'2d. Selected sticks reasonably straight and air-sea- 
soned under cover for two years and over. 

"3d. Average sticks cut from lumber which had been 
in open-air service for four years and over." 

If /= length of column in inches, 

d = least side of column section in inches, 
and p = Ult. Comp. resistance in lbs. per sq. in. ; 

then the formulae found for these three groups were : 
ForNo.i: /.^-^^^^ 



I P' 



2Sod' 

Ty AT S2OO 

For No. 2 : p = ^ 

300 a' 

For No. 3 : ^ = ^ 



I P' 
2Sod^ ■ 



But in order to provide against ordinary deterioration 
while in use, as well as the devices of unscrupulous builders, 
Mr. Smith recommends the formula for group No. 3 as the 
proper one for general application. He also recommended 



J 



that the factor of safety be ^/- until 25 diameters are 

reached, and five thenceforward up to 60 diameters. 
This last limit he regards as the extreme for good 
practice. 



Art. 86.] 



TIMBER COLUMNS, 



533 



Tests of White Pine and Yellow Pine Full-size Sticks with 

Flat Ends. 

In consequence of the usual manner of simply abutting 
the end of timber columns against their supports, all such 
members are practically always assumed to have flat ends, 
but this expression does not mean accurately squared '' flat 
ends." Tables III and IV have been formed by digesting 
the results of tests of nearly or quite full-size white and 
yellow pine timber columns made at the U. S. Arsenal at 
Watertown, Mass., and reported in " Ex. Doc. No. i, 47th 
Congress, 2d Session," constituting one of the best series 
of timber column tests yet made in this country. 

Each result in both Tables is usually a mean of from 
two to four tests, although a few belong to one test only. 
All timber, both of yellow and white pine, was ordinary 
merchantable material, with about the usual defects, knots, 
etc., and failure frequently took place at the latter ; it was all 
well seasoned, and all columns were tested with flat ends. 

Table III. 
YKLLOW-PINE COLUMNS WITH FLAT ENDS. 



Length. 


Size of Stick 
Inches. 


1' 


Ultimate 
Compres- 
sive Re- 
sistance, 
Lbs. per 
Sq. In. 


Length. 


Size of Stick, 
Inches. 


/ 
d' 


Ultimate 
Compres- 
sive Re- 
sistance, 
Lbs. per 
Sq. In. 


Ft. Ins. 








Ft. 


Ins. 








15 


8.25X16.25 


21.7 


3,445 


15 





5-oXi2 


35.6 


3,764 
3,304 


10 

16 8 


5-5 X 5.5 


22 


4,738 


23 


4 


7.7X 9-7 


36.4 


7.7 X 9-7 
6.6 XI5.6 


26.7 


4,384 


17 


6 


5.5X 5-5 


38.2 


3,242 


15 


27.0 


3,593 


15 





4.5X11.6 


41 


2,462 


12 6 


5-5 X 5-5 


27.3 


5,077 


26 


8 


7-4X 9-4 


43 


2,893 


15 


5-9 X12.0 


30.8 


3,546 


15 





4.0X1 1.4 


44 


3,065 


20 


7.6 X 9-6 


31-2 


3,496 


20 





5.4X 5.4 


44-3 


2,867 


15 


5.7 X11.7 


31.9 


3,106 


22 


6 


5.5X 5-5 


50 


2,065 


15 


5.6 XI5-6 


32.1 


3.656 


25 





5.5X 5.5 


55 


1,856 


15 


5.5 X 5.5 


32.8 


3,962 


27 


6 


5.3X 5-3 


62.3 


1,709 



534 LONG COLUMNS [Ch. X. 

Flat-end yellow-pine columns were observed to begin to 
fail with deflection at a length of about 2 2d, d being the 
width or least dimension of the normal cross-section. All 
columns were of rectangular section, and / in the following 
table is the length. Table III, therefore, includes no short 
column, i.e., one which failed by compression alone with 
no deflection. 

About sixteen of the latter were tested with the follow- 
ing results: 

CM, ^ 11 • 1 ( maximum = 5,677 lbs, per sq. in. 

Snort yellow-pme columns ; J _, ' << << ^« 

7 ju 1 M rnean =4,442 

L-^d below 22 ) ■ ■ ^'^^ ,, <( ,( 

( mimmum =3,430 " 

Each of the preceding tests was made on a single rectan- 
gular stick. A number of tests, however, were made on 
compound columns formed by bolting together from two 
to three rectangular sticks, with bolts and packing or 
separating blocks at the two ends and at the centre. The 
bolts were parallel to the smaller sectional dimensions of 
the component sticks. As was to be expected, those 
compound columns possessed essentially the same ultimate 
resistance per square inch as each component stick con- 
sidered as a column by itself, as the following results show. 
/ is the length of the column and d the smallest dimension 
or width of one member of the composite column. All 
had fiat ends. 

l-^d. Number of Tests. 

C maximum = 4,559 lbs. per sq. in. 
32.1 18 ^ mean =3,841 

(minimum =2,756 

(maximum = 3,*357 
36 18.. i mean =3,122 

(minimum =2,942 

Table IV gives the results for white-pine columns, and 
corresponds with Table III, in that it shows only the failures 
with deflection, which was observed to begin with those 
columns at a length of 32^/. / and d possess the same 



Art. 86.] 



TIMBER COLUMNS. 



535 



Table IV. 

WHITE-PINE COLUMNS WITH FLAT ENDS. 









Ultimate 


1 








Ultimate 








Compres- 


1 ■ 








Compres- 




Size of Stick, 


I 


sive Re- 






Size of Stick, 


/ 


sive Re- 


Length. 


Inches. 


d ' 


sistance, 




Inches. 


d ' 


sistance, 








Lbs. per 


i 








Lbs. per 








Sq. In. 


1 
1 








Sq. In. 


Ft. Ins. 








1 Ft. 


Ins. 








15 


5-6XI5-6 


32 


1,874 


17 


6 


5-4X5.4 


40 


1. 841 


20 3 


7-4X 9-3 


32.4 


2,448 


26 


8 


7.5X9-3 


42.7 


2,113 


15 


5-6XII.5 


32.7 


2,432 


20 





5.3X5.3 


45 


1,455 


15 3 


5-4X 5.4 


33 


2,744 


22 


6 


5-2X5-2 


52 


1,501 


23 4 


7-7X 9-6 


36.4 


2,072 


25 





5-3X5.3 


57 . 


952 


15 


4-5X11.6 


40 


1,672 


27 


6 


5.4X5.4 


62 


1,081 



signification as in Table III, the column l^d showing 
the ratios between the lengths and least widths. 

Thirty columns with lengths less than 32(i were tested 
to destruction. These sticks failed generally at knots by 
direct compression and without deflection. The results 
of these thirty tests were as follows: 



Short white-pine columns ; 
/-r-c^ below 32 



All the preceding white-pine columns were single sticks, 
but a large number of built posts composed of two to four 
white-pine sticks bolted together, with spacing blocks at the 
two ends and at the centre, were also tested with the results 
shown below, l^d is the ratio of length over least width of 
a single stick of the set forming the composite column. 

l^d. Number of Tests. 



maximum 


= 3 


700 


lbs. 


per sq. 


in. 


mean 


= 2 


414 


" 


" 


' 


minimum 


= 1 


687 


" 


" 


* 



maximum 



32.1 



15. 



C maxii 
< mean 
( minin 



2,273 lbs. per sq. in. 
i,q8o 



minimum =t,66i 

) maximum = 2,255 
mean =1,099 

minimum =1,804 



AO. 



maximum 


= 2 


,021 


mean 


= 1 


,830 


minimum 


= I 


,419 



536 



LONG COLUMNS. 



[Ch. X. 



A comparison of these results with those given in Table 
IV shows that these composite or built columns were the 
same in strength per square inch with the single sticks 
of which they were composed, the latter being considered 
single columns. 

All the white-pine composite columns were tested witl: 



Plate F. 












1 1 1 1 M 1 111; ! 1 1 ! r ! 11 ' MM M 1 ' M i 1 M M ' M 1 MM 1 M . 1 ' " 


III! II 1 MM MM 1 1 11 • 1 1 1 1 MM \ \ \ 1 1 M 11 1 II 11 1 1 




III 1 L-j 1 1 Tf " "TtrT^'^^ji: — H 








IIWU ' |Wni e. r^ii-iit| o,i,iwr\o, | || {|| ||| M i T ■■'*-.^ ^ -"■- 






i M 1 1 1 1 1 1 1 Ml 1 1 M ■ II ■ 1 1 M M M 1 1 M 






III 1 1 1 M M M 1 1 1 1 M 1 M 1 1 1 M M 1 1 1 




III 1 M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M 1 1 1 




M 1 Mill MM' ' M Mill 1 M M ' 


i 1 I ! M Ml 1 1 ! 




M 1 III j 1 1 M 1 1 1 M 1 1 11 M 1 1 


VUY\ V a 1 1 jl 11 M 1 1 1 1 1 1 M 1 1 1 M 1 1 ; 1 1 


5000 Ids- 1 i m 1 m i ' " M 


1 M 1 1 1 1 II 1 1 M M 1 1 1 M 1 1 M M 








WW H rf^ ""-..^ 1 1 










3000 ------- - 1 ^^, '1 ■ 


i^iJ 






glmn • ""V1l-iJO' W-Pl N-E SI"" -lirxS " -**«*. '' -~- 




^:: :::::: ::::-Wl:jm^%L---------------- ------------ --»^-^-^J^-^-;j 




lUUO--' --^ -..._._.... I ; 




h -- -- ^ - - 1 ' ' i ■ - - - - j J 






llm.!' ^P 20 1 1 ^M 401 1 f- 1 SO |^_ ^ I _£ 



flat ends and were built up with the greatest widths of 
individual sticks adjacent to each other. 

The results in Tables III and IV are shown graphically 
in Plate F. One ordinate gives the values of l-^d, and the 
other the ultimate resistance in pounds per sq. in. 

The full curved lines running into horizontal tangents at 
the left represent about mean lines through the points 
indicating the actual column tests. 

The broken lines represent the following empirical for- 
mulae ; in which p is either the ultimate resistance or work 
mg stress in pounds per sq. in. 



Art. 86.] TIMBER COLUMNS. 537 

For yellow pine . . . _/? = 5800— 7o(/-f-(i) 
" white " ... p = ^Soo - 4j (l -^ d) 

For wooden railway structures there may be used: 

For yellow pine . . . ^ = 750 — 9(/-i- J) 
'' white "... p = soo-6{l^d) 

For temporary structures, such as bridge false works 
carrying no traffic: 

For yellow pine . . . p = i$oo—i%{l^d) 
'' white *' . . . p = iooQ-i2{l-rd) 

The preceding formulce are to be used only between the 

limits oj - = 20 and -=60 for yellow pine and ■-}=3o and 
da a 

- = 60 for white pine, 
d 

For short columns below - = 20 and -=30 there are to 

d d 

be used for yellow and white pine respectively : 

Ultimate. Railway Bridges. Stm?tu?e7 

Yellow pine. . . .^ = 4400 550 iioo lbs. per sq. in. 

White " . . . .p = 2400 300 600 " " " 

All the preceding values are applicable to good average 
lumber for the engineering purposes indicated. 

Table V exhibits a number of results of the tests of 
short timber columns taken from the '* U. S. Reports of 
Tests of Metals and Other Materials" for 1894, 1896, 1897, 
and 1900. It will be observed that the ratios of length over 
thickness, i.e., minimum dimension of cross-section, are 
less than 22, and with two exceptions much less. These 
columns do not, therefore, come within the range of appli- 
cation of such formulae as those given on the preceding 
page for yellow pine and white pine. 



538 



LONG COLUMNS. 



ich. X. 



Table V. 

SHORT TIMBER COLUMNS. 



Timber. 



Long-leaf pine.. 
Short-leaf pine. 



Spruce. ....... 

Long-leaf pine * 

Cypress 

White pine. . . . 



Red oak. . . 
Douglas fir. 



White oak . 



Dimensions, I 


iches. 










/. 












Thick- 


d 


Leng'h 


Bre'dth 


ness. 




1 20 


9.8 


9.8 


I 2 


120 


9.8 


9.8 


12 


120 


9.8 


9.8 


12 


120 


9.8 


9.8 


12 


120 


9.8 


9.8 


12 


120 


9.8 


9.8 


12 


120 


9.6 


9.6 


12 


131 


9-S 


9-5 


14 


120 


8 


8 




58 


9-5 


7-9 


8 


71 


9-5 


7-9 


9 


48 


4 to 6 


3-5 


14 


60 


14 


2.8 


22 


60 


8& 14 


3 


20 


60 


12 


4.1 


15 


121 


10 


8 


IS 


106 


10&12 


10 


1 1 


74 


7-5 


7-9 


lo 


6q 


10 


8 


9 



Ultimate Compressive 

Resistance, 

Lbs. per Sq. In. 



Max. Mean. Min. 



4,976 

3,800 
4,200 
3,925 
3,400 
4,000 
3,174 
7,354 
3,457 



6,247 



4,574 
3,558 
3,957 
3,481 
^,000 
3,568 
2,589 
6,093 
3,308 
3,652 
2,917 
5,160 
6,211 
6,725 
6,220 
3,697 
4,214 
4,372 
4,042 



4,200 
3,369 
3,714 
3,037 
2,600 
3,135 
1,900 
4,960 
3,113 



2,917 
S>568 



Butt sticks. 

Top 

Middle " 

Butt 

Top " 

iViiddle 

Old posts. 

Probably 
170 years 
old. 



* Well seasoned and dry; 12 years old. Had been in a fire and corners were partially 
charred. 

All posts represented in this table contained probably 1 5 to 18 per cent, of moisture, or 
perhaps more. 

The long-leaf and short-leaf pine tests show that columns 
taken from the butts of trees are stronger than those 
taken either from the middle or the tops, the top sticks, 
as a rule, having the least ultimate resistance per square 
inch of all. The white -pine and red-oak sticks yield interest- 
ing results on account of their age, as they were taken from 
some wooden trusses of the Old South Church, Boston, 
Mass., a building constructed in 1729. The timber was so 
housed as to be completely protected and kept very. dry. 
The results show no loss of resistance as compared with 
tests of the same kind of timber at the present time. 

The effect of immersion in water on the resistance of 
timber is illustrated by tests made at the Watertown 
Arsenal. A post similar to one of the old long-leaf pine 
columns, 12 of which were tested in a seasoned condition 



Art. 86.] TIMBER COLUMNS. . 539 

giving the average shown in the Table of 6093 pounds per 
square inch, was submerged in water for a period of 130 
days and then tested with the result of failing at 3800 
pounds per square inch. 

The values given in Table V correspond closely to the 
results shown for yellow pine and white pine on pages 
534 and 535, so far as they may properly be compared. 



CHAPTER XL 

SHEARING AND TORSION. 

Art. 87. — Modulus of Elasticity. 

It has already been shown in some of the Articles of 
the first part of this book that the stresses of shearing and 
torsion are identical, both being shears; hence the modulus 
of elasticity is the same for both. 

As it is much more convenient to make accurate deter- 
minations of the modulus of elasticity in torsion than in 
direct shearing, the former method has been employed in 
practically all cases. A number of such moduli for four 
varieties of steel are given in Art. ^S. Those values show 
that the modulus changes but little for the different varieties 
of steel indicated. 

The aggregate of torsion tests so far as they have been 
made indicate that the two moduli of elasticity, G for shear 
and E for direct stresses of tension and compression, have 
the approximate relation : 

G = {.4 to .45)^- 

Prof. Bauschinger published in *' Der Civilingenieur, " 
Heft 2, 1 88 1, the results of some of his tests of cast-iron 
cylinders or prisms which are still valuable on account of 
the accuracy with which he made his determinations. 
The prisms were about 40 inches long, and were subjected 
to torsion, while the twisting of two sections about 20 inches 

540 



Art. 87.] 



MODULUS OF ELASTICITY, 



541 



apart, in reference to each other, was carefully observed. 
The results for four different cross-sections will be given — 
i.e., circular, square, elliptical (the greater axis was twice the 
less), and rectangular (the greater side was twice the less). 
In each case the area of cross-section was about 7.75 square 
inches. The angle a. is the angle of torsion — i.e., the 
angle twisted or turned through by a longitudinal fibre 
whose length is unity and which is at unit's distance from 
the axis of the bar. 



Section. 



Hrrnlar j 0.007 degree 7,466,000 lbs. per sq 

^^^^^^^^ 1^^-^ - 6,157,000" " 

7,437,000 " 

6,228,000 " 

7,039,000 " 

5,987,000 " 



Elliptical...... ..]°;°°9 

Square 

Rectangular 



IC 



076 
0.008 



073 
o.oi 
0.08 



6,996,000 
5,716,000 



The formula by which G is computed, when the torsional 
moment and angle a are given, is the following: 



G = 



M I^ 

'A 



a 



(i) 



in which M is the twisting moment, A the area of the cross- 
section, Ip the polar moment of inertia of that cross-section, 
and c a coefficient which has the following valuep 

47:^ = 39.48 for circle and ellipse, 
42.70 *' square, 
42.00 " rectangle, 

as shown in Appendix I. 

Bauschinger's experiments show that the coefficient of 
shearing elasticity for cast iron may be taken from 6,000,000 
to 7,000,000 pounds per square inch; also that it varies for 
different ratios between stress and strain. 

It has been shown in Art. 6, that if E is the coefficient 
of elasticity for direct stress, and r the ratio between direct 



m 



542 SHEARING AND TORSION. [Ch. XI. 

and lateral strains, for tension and compression, that G 
may have the following value: 

E 

Prof. Bauschinger, in the experiments just mentioned, 
measured the direct strain for a length of about 4 inches, 
and the accompanying lateral strain along the greater axis 
of the elHptical and rectangular cross-sections, and thus 
determined the ratio r between the direct and lateral strains 
per imit in each direction. The following were the results: 

Compression. 
Section. r. G. 

Circular 0.22 6,541,000 lbs. per sq. in 

Elliptical 0.23 6,541,000 " " " 

Square ..0.24 6,442,000 " " " 

Rectangular 0.24 6,499,000 " " " 

TENSION. 

Circular o. 23 6,570,000 lbs. per sq. in. 

Elliptical 0.21 6,811,000 " " " 

Square 0.26 6,399,000 " " " 

Rectangular 0.22 6,527,000 " " " 

The values of E are not reproduced, but they can be 
calculated indirectly from eq. (2) if desired. 

It is seen that the values of G, as determined by the 
different methods, agree in a very satisfactory manner, 
and thus furnish experimental confirmation of the funda- 
mental equations of the mathematical theory of elasticity 
in solid bodies. 

The fact that G is essentially the same for all sections is 
also strongly confirmatory of the theory of torsion in 
particular. 

These experiments show^ that, for cast iron, the lateral 
strains are a little less than one quarter of the direct strains. 
If r were one quarter, then G =|-E, or E =|G^. 



Art. 88.1 ULTIMATE RESISTANCE. 543 

Art. 88. — Ultimate Resistance. 

It has seemed more convenient to give some values of 
ultimate and working resistances for the materials iron and 
steel which are much more commonly used than any others 
to resist torsion in Arts. 37 and 38, where the complete 
analyses of the formulae for the common theory of torsion 
are given. Those articles should, therefore, be consulted 
for such formulae and analytic operations as are involved 
in the design of shafting to resist torsion. The experimental 
values set forth in the following articles may be employed 
in the formulae of the common theory of torsion for any de- 
sired practical operation in the design of torsion members. 

Before considering the ultimate shearing resistance of 
special materials it will be well to notice the two different 
methods in which a piece may be ruptured by shearing. 

If the dimensions of the piece in which the shearing force 
or stress acts are very small, i.e., if the piece is very thin, 
•the case is said to be that of "simultaneous" shearing. If 
the piece is thick, so that those portions near the jaws of 
the shear begin to be separated before those at some dis- 
tance from it, the case is said to be that of "shearing in 
detail." In the latter case failure extends gradually, and 
in the former takes place simultaneously over the surface 
of separation. Other things being the same, the latter 
case (shearing in detail) , will give the least ultimate shearing 
resistance per unit of the whole surface. 

In reality no plate used by the engineer is so thin that 
the shearing is absolutely simultaneous, though in many 
cases it may be essentially so. 

Wrought Iron. 

There may be found in the Articles on Riveted Joints 
some experimental determinations of the ultimate shearing 



544 SHEARING AND TORSION. [Ch. XI. 

resistance of wrought iron which, under the conditions of 
such joints, may range from about 34,000 to about 43,000 
pounds per square inch. It has been observed in the 
consideration of riveted joints that the ultimate resistance 
to shear of rivets will generally be less with thick plates 
than with thin, because the bending stresses of tension 
and compression will generally be greater for thick plates 
than for those that are thinner. If the riveted joint is so 
designed that the bending stresses are not greater for thick 
plates than for thin ones, the effects of bending will neces- 
sarily disappear. 

Such tests as have been made on direct shearing resist- 
ance show that generally it may safely be taken at 35,000 
to 40,000 pounds per square inch, or if S is the ultimate 
shear per square inch and T the ultimate tensile resistance 
of wrought iron per square inch, there may be taken ap- 
proximately 

Cast Iron. 

There are few tests available for the determination of 
the ultimate shearing resistance of cast iron. For the ordi- 
nary grades, such as cast-iron water pipes and similar soft 
gray -iron castings, the ultimate shearing resistance • has 
sometimes been taken equal to the ultimate tensile resist- 
ance, i.e., 15,000 to 18,000 pounds per square inch, but 
this is probably too high except for the special stronger 
grades of material. 

For general purposes it is probably safe to take the ulti- 
mate shearing resistance of cast iron about three-quarters 
of its ultimate tensile re^stance. It should only be used 
for shearing, however, at a low working stress, depending 
obviously on the. purpose for which its use is contemplated. 



Art. 88.] ULTIMATE RESISTANCE. 545 

Steel. 

The results of Prof. Ricketts' shearing tests on both open- 
hearth and Bessemer steel rounds with different grades of 
carbon are given in Table I of Art. 43. The elastic limit 
is the point at which the metal first fails to sustain the scale 
beam. The double-shear resistance in one case exceeds the 
single by over six per cent. According to these tests, the 
ultimate shearing resistance of mild steel may be taken 
at three quarters of its ultimate tensile resistance. Each 
shear result is a mean of three tests. 

The rivet steel was low, containing but .09 per cent, of car- 
bon. While the specimens of Bessemer steel were a little 
higher in carbon, ranging from . 1 1 to . 1 7 per cent., except the 
last six, they were also of low or medium steel. It should 
be carefully noted that the results in that table show that 
the ultimate' shearing resistances for the low or medium 
steels running from 44,600 pounds per square inch up to 
53,260 pounds per square inch are closely three fourths 
the corresj^onding ultimate tensile resistances. On the 
other hand, the six specimens of high steel give ultimate 
shearing resistances but little over two thirds of the corre- 
sponding ultimate tensile resistances. This is a feature of 
the relation between the ultimate shearing and ultimate 
tensile resistances of different grades of steel which is 
commonly exhibited in tests. The high steel appears to 
yield an ultimate shearing resistance of sensibly less per- 
centage of the tensile ultimate than low steel. 

In the Arts. 74 and 76 on riveted joints there will be 
found a number of values of ultimate resistance for steel 
rivets in shear. They constitute important determinations 
of the ultimate shearing resistance of steel rivets under con- 
ditions in which they are frequently used. 



546 



SHEARING AND TORSION. 



[Ch. XI. 



Copper, Tin, Zinc, and Their Alloys. 

The following values of the ultimate resistance to torsive 
shear Tm, were determined by Prof. R. H. Thurston in his 
early experimental work on the bronzes. Although these 
determinations were made on test specimens only .625 inch 
in diameter and with a torsion length of i inch, they con- 
stitute practically the only fairly complete shear and torsion 
data on the copper-tin and copper-zinc alloys. 

Table I. 



Composition. 


Ultimate Torsive 
Shear, T,n- 


Elastic Limit, 
PerCentof r,„. 


Ultimate Torsion 






Angle. 


Cu. 


Sn. 












Pounds. 




Degrees. 


100 


00 


35,910 


35 


1530 


100 


<X) 


28,430 


40 


52 to 154 


00 


100 


3,196 


45 


557-0 


00 


100 


3,297 


33 


691 .0 


90 


10 


43,943 


41 


II4-5 


80 


20 


47,671 


62 


16.3 


70 


30 


4,407 


100 


1-5 


62 


38 


1,770 


100 


1 .0 


52 


48 


686 


100 


I.O 


39 


61 


5,881 


100 


1-7 


29 


71 


5,257 


100 


2.34 


10 


90 


5,761 


63 


131.8 


90 


10 


25,027 


49 


57-2 


90 


10 


31,851 


57 


72.6 



Tm is in pounds per square inch. 



Table I relates to alloys of copper and tin, and Table II 
to alloys of copper and zinc. 

None but specimens with circular sections were tested. 

An examination of the results given in Tables I and II 
show that the resisting capacities of each series of alloys 
vary greatly with the varying elements constituting the 
alloy. Indeed, the shearing resistances of these alloys in 
torsion are seen to vary as widely as their tensile resistances. 



Art. 88.1 



ULTIMATE RESISTANCE. 



547 



Table II. 



Percentage of 












Ultimate Torsive 
Shear, T,, 


Elastic Limit, 
Per Cent of r^,^. 


Ultimate Torsion 






Angle. 


Copper. 


Zinc. 












Pounds. 




Degrees. 


90.56 


9.42 


35.100 


17.2 


458.0 


81.90 


17.99 


41,575 


27-5 


345 -o 


71 .20 


28.54 


41,000 


24.0 


269.0 


60.94 


3H-65 


48,520 


29.4 


202 .0 


55-15 


44-44 


52,320 


32.7 


109.0 


49.66 


50.14 


43,154 


36.0 


38.0 


41 -30 


58.12 


4,5^8 


100. 


1.8 


32.94 


66.23 


7,241 


100. 


1.2 


20.81 


77-63 


16,374 


100. 


0.8 


10.30 


88.88 


22,500 


85.6 


71 


00 


100.00 


9,186 


38.1 


141. 5 



Although the values of Tm are the ultimate intensities 
of torsive shear, they may be accepted as ultimate resist- 
ances for direct shear for the same alloys. 



Timber. 

The shearing resistance of timber is least along planes 
parallel to the fibres and greatest when the shearing force 
acts in planes at right angles to the fibres. Again, the 
shearing resistance parallel to the fibres is somewhat differ- 
ent in short blocks from that found in full-size beams sub- 
jected to flexure. In the latter case it has been shown in 
Art. 15 that the greatest intensity of shearing stre s 
parallel to the fibres will take place in the neutral surface. 
It has been found that for relatively short spans timber 
beams in flexure will fail by shear along the neutral surface. 
Hence the ultimate resistance to shear along that surface 
has much practical value and it has been determined in 
tests of many full-size beams. Among the latter those made 
by Prof. Arthur N. Talbot at the University of Illinois and 



548 



SHEARING AND TORSION. 



[Ch. XL 



described in the University of Illinois Bulletin No. 15, 
December, 1909, are of unusual value. The full-size beams 
were 13.5 feet to about 14.5 feet span and with cross-sections 
of 7 inches by 12 inches, 7 inches by 14 inches and 7 inches 
by 16 inches. Other smaller beams were, however, used. 
The beams were of sound merchantable lumber and of about 
the quality used in good engineering work. The following 
table gives the results of these tests, showing the number 
of pieces tested to failure with the highest, average and lowest 
ultimate shear per square inch along the fibres in or near 
the neutral surface. 



Table III. 

ULTIMATE RESISTANXES ARE GIVEN IX 
SQUARE INCH 



POUNDS PER 



Timber. 



No. of 

Pieces. 



Untreated longleaf pine . 
Untreated shortleaf pine. 
Creosoted shortleaf pine. 
Creosoted loblolly pine. . 
Untreated loblolly pine. . 
Old Douglas fir. ....... . 

New Douglas fir 



25 
4 
6 
8 

10 
10 
II 



Ultimate Shearing Stress. 



Highest. Average. Lowest 



497 
505 
410 

391 

388 

383 
401 



370 
364 
302 

273 
314 
298 

323 



188 

293 
224 
224 

253 
221 

275 



The values given in Table III are somewhat smaller 
than those which Prof. Talbot found for short blocks. The 
ultimate shearing stress along the fibres of the neutral 
axis of the full-size beams ranged from 75 per cent, up to 
loi per cent, of the corresponding results for short blocks 
of the same kind of timber. The new Douglas fir gave the 
highest of these percentages and untreated longleaf pine 
together with creosoted loblolly pine gave the smallest. 

The American Railway Engineering and Maintenance of 
Way Association- has recommended ultimate and working 



Art. 88.] 



ULTIMATE RESISTANCE, 



549 



values for shear along the grain and in the neutral surface 
of beams as given in Table IV of Art. 90. 



Natural Stones. 

The ultimate shearing resistance of stones has not as 
great practical value as the ultimate compressive or the 
ultimate bending resistance, yet there are occasional 
structural conditions under which it is necessary to ascer- 
tain what shearing capacity may be relied upon. Valuable 
data for this piu-pose are shown in Table IV taken from the 
•' U. S. Report of Tests of Metals and Other Materials" for 
1894 and 1899. The sheared surfaces were about 6 inches 
by 4 inches in area. Generally one such surface was 
sheared, but occasionally two. 

Table IV. 
SHEARING RESISTANCE OF NATURAL STONES. 



Stone. 


Ultimate Shearing Resistance, Lbs. 
per Square Inch. 




Maximum. 


Mean. 


Minimum. 


T-5randford oranite Conn 


1,925 

2,872 
2,231 


1,834 
2,554 
2,219 
1,825 

1,549 
1,369 
1,237 
1,411 
1,242 
1,332 
1,490 
1,705 
1,831 
1,204 
1,150 
1,243 
1,352 
2,127 


1,742 
2,236 


Milford Granite, Mass 


Troy granite N H 


2,197 


Milford pink granite Mass 




Pi aeon Hill crj-anite Mass 


2,047 

1,501 

1,554 
2,016 

1,287 
1,308 
1,383 

2,518 


1,052 


Creole marble, Ga 


Cherokee marble Ga 




Etowah marble Ga 




KennesaAv marble, Ga 

Marble Hill marble, Ga 

Tiickahoe marble NY 


1,163 
1,426 


Mount Vernon limestone Ky 


i,"^8q 


Cooner sandstone Ores^on 




Mavnard sand'^tone Mass 


1,120 


Xibbe sandstone Alass 


992 




1,102 






Vcimmp>rtha1 1imf>«;tnTlP T^llTTalo . . . 


1,735 





55° 



SHEARING AND TORSION. 



[Ch. XL 



All the results except the last are taken from the Report 
for i8q4. Where but one value appears in the table one 
test only was made. In the other cases two tests were 
made and the mean values are means of the two shown in 
the columns containing the greatest and least. It will be 
observed that the ultimate shearing resistance is scarcely 
more than ten per cent, of the ultimate compressive re- 
sistance of the various stones tested. 

The greatest permissible working stresses for natural 
stones in shear, in design work, will necessarily depend 
on the duty to be performed. In view of the variable char- 
acter of even the best of natural stones as delivered ready 
for use, one eighth to one tenth of the ultimate is as much as 
should be taken in ordinary cases, and materially less than that 
under some conditions. 

Bricks. 

The shearing resistance of bricks, like that of natural 
stones, is seldom employed, but it is sometimes needed. 
The ultimate resistances of bricks in shearing shown in 
Table V are taken from the " U. S. Report of Tests of 
Metals and Other Materials " for 1894. 

Table V. 

BRICKS IN SHEARING. 



Kind of Brick. 



R--'- 'sheared 
Sur- 



ance. Lbs. 

per Square 

Inch. 



faces. 



Hydraulic Press Brick Co., St. Louis, No. 6 

" 511 

" " " " " brown 

" " " " Chicago, red 

Northern Hydraulic Press Brick Co., Minneapolis, dark red. 
Eastern " " " " Philadelphia, 210. . . . 

" " " " " " 220 

" ." " 390 

Philadelphia and Boston Face Brick Co., Boston, gray 



1,011 
642 

1,047 
784 
714 

1,167 

1,097 
988 
433 
639 



1 



Art. 88.J ULTIMATE RESISTANCE. 551 

In these shearing tests the sheared surfaces were each 
about 2.25 by 4 inches in dimensions. 

The ultimate shearing resistances in Table V range 
scarcely 10 to 20 per cent, of the ultimate compressive resist- 
ances of the same materials shown in Art. 68. 

Working shearing stresses for design operations should 
not be taken more than one eighth to one tenth of the 
ultimate values found in Table V, 



CHAPTER XII. 

BENDING OR FLEXURE. 

Art. 89. — Modulus of Elasticity. 

The modulus of elasticity as determined by experiments 
in flexure can scarcely be considered other than a con- 
ventional quantity. If the span of a beam were very long 
compared with the depth of the beam and if the moduli 
of elasticity for tension and compression were equal to each 
other, and if all the hypotheses involved in the common 
theory of flexure were true, then the modulus of elasticity 
for flexure would be a real quantity and essentially the same, 
at least, as that for either tension or compression. 

These conditions, however, do not exist in bent beams 
and the quantity ordinarily called the modulus of elas- 
ticity in flexure possesses value chiefly as an empirical 
factor which enables deflection, independently of shear, to 
be estimated with sufficient accuracy for all usual purposes. 

The formulas to be employed in the determination of 
the modulus of elasticity for flexure have already been 
established in connection with the common theory of flexure 
and their use will be shown in succeeding articles. 

Art. 90. — FormulaB for Rupture. 

The formula of the common theory of flexure, available 
for practical use, are true only within the limits of elas- 
ticity. In the testing of beams to failure they are employed 
precisely as if the elastic properties of the material were 
maintained up to the degree of loading which causes failure. 

552 



Art. 90.] FORMULA FOR RUPTURE. 553 

While this, strictly speaking, is irrational, it is the only 
satisfactory procedure available. By placing the analytic 
expression for the moment of the internal stresses in the 
normal section of a bent beam equal to the moment of the 
external loading causing failure, the resulting equation may 
be solved so as to give the apparent ultimate intensity of 
stress k in the extreme fibres of the beam. The so- 
called intensity of fibre stress found in this manner is an 
empirical quantity which may be introduced into the for- 
mulae of the common theory of flexure and so make them 
applicable to the operations of engineering practice in con- 
nection with loaded beams of any shape of cross-section. 

If k and k'^ are the greatest intensities of stress in the 
section of rupture and at the instant of rupture; y the 
variable normal distance of any fibre from the neutral sur- 
face; yi and y^ the greatest values of y; b the variable 
width of the section (normal to y) ; and M the resisting 
moment at the instant of rupture ; then the general for- 
mula for rupture by bending, as given by eq. (i) of Art. 
26, is 

- I y'^hdy-^ — -, \ y^bdy. . . . (i) 

'1> y J-y' 

This equation is in reality based on the supposition that 
the moduli of elasticity for tension and compression are not 
equal. It is rare, however, that such a supposition is made. 
It is practically the invariable rule to assume the moduli 
of elasticity for tension and compression to have equal 
values and such an assumption is fortunately sufficiently 
accurate for all ordinary purposes. 

If the tensile and compressive moduli of elasticity are 

k k' 
the same — =—. and eq. (i) becomes 

yi y 

^=57 ,• • (^) 



M = 
y^ 



554 BENDING OR FLEXURE. [Ch. XII. 

This is the usual equation of flexure employed so fre- 
quently in connection with the design of bent beams or the 
investigation of their carrying capacity, I being the moment 
of inertia of the normal section of the beam di the distance 
of the most remote fibre from the neutral axis of the section 
and M the moment of the external forces or loading about 
the neutral axis of the section in question. In the prac- 
tical use of this formula it is only necessary to introduce the 
proper values of I and di for the shape of a section involved. 

Art. 91. — Beams with Rectangular and Circular Sections. 

These are the simplest forms of sections for bent beams 
employed in engineering work. Timber beams are with few 
exceptions of rectangular section and so are many rein- 
forced concrete beams, although in such a case the section 
is composite, i.e., composed of two materials, and it will 
receive separate treatment in a later article. The solid 
circular section belongs to pins in pin-connected truss 
bridges whose design always involves their consideration 
as a loaded beam of very short span. 

The following are the values of / and d^ for rectangular 
and circular sections, h being the side of the rectangle normal 
and h that parallel to the neutral axis, while r is the radius 
of the circular section and A the area in each case : 

... (i) 





bh' Ah' 




12 12 


Rectangular: - 


2 




r , Tzr' Ar'' 




/ = — =— -, 


Circular: < 


4 4 




^d,=r. 



(la) 



Art. 91.] BEAMS WITH RECTANGULAR AND CIRCULAR SECTIONS. 555 

If the beams are supported at each end and loaded by a 
weight W at the centre of the span (or distance between 
supports) , which may be represented by /, then the moment 
at the centre of the beam becomes 

JP:r=M=— (2) 

4 

There will then result from eq. (2), Art. 89: 
For rectangular sections: 

,. Wl khh? kAh , , 

M=— -=-—=-— . ..... (3) 

400 

For circular sections : 

.. Wl irkr^ kAr , , 

M= — = = (4) 

4 4 4 

The quantity k is called the modulus of rupture for 
bending, and if experiments have been made, so that W 
is known, eq. (3) gives 

3M^3]^ 

2 Ak 2 bh^' ^^^ 

and eq. (4) 

r^ Wl Wl ,^, 

^=^-=— ^. ....... (6) 

Ar xr ■ 

If the rectangular section is square, bk^ =b^ =h^. 

Steel. 

If the beam is simply supported at each end and carries 
a load W at the centre, while E is the coefficient of elasticity 
and w the deflection at the centre, eq. (28) of Art. 28 gives 

Wl' / ^ 

" = 4^- .^'^ 



556 BENDING OR FLEXURE. [Ch. XII. 

If, in any given experiment, w is measured, E may then 
be found by the following form of eq. (7) : 

-(8) 



4SWI 

If the section is rectangular 
^ WP 



4wbh' 



(9) 



These equations enable the coefficient of elasticity E to 
be computed readily from experimental observations. It 
IS only necessary to measure accurately the deflection w 
produced by the load or weight IF and then introduce all the 
known quantities in eq. (8) or eq. (9). 

A bar of wTought iron 3 inches deep and i inch wide 
was placed on supports 48 inches apart and loaded with a 
weight of 400 pounds at mid-span. The measured de- 
flection was .0138 inch. Hence 

J-, 400X48X48X48 

E = — - = 29,730,000. 

4X1 X3X3X3X. 0138 

Other applications may be made in precisely the 
same way. 

High Extreme Fibre Stress in Short Solid Beams. 

During the period when wrought iron was used for 
structural purposes, especially for wrought-iron pins with 
diameters up to 9 or 10 inches, it was observed that if the 
ultimate extreme fibre intensity k was computed by eq. (5) 
or (6) with data obtained by actual test, the result would 
be excessively high, i.e., far beyond the ultimate resistance 
to tension. These pins, however, on which are packed the 



Art. 91.] BEAMS WITH RECTANGULAR AND CIRCULAR SECTIONS. 557 

lower chord eye-bars of an ordinary truss bridge, have very 
short spans, indeed the span is usually much less than the 
diameter of the pin and sometimes less than one quarter 
of the diameter of the pin. It should be remembered in 
this connection that the common theory of flexure is im- 
plicitly if not explicitly based upon the condition that the 
length of span of the bent beam must be long compared with 
the depth of the beam. In fact the span should be many 
times that depth, and the longer it is the more nearly 
correct becomes the common theory of flexure. These ob- 
servations are equally true whether the cross-section of the 
beam_ is circular or rectangular or has any other shape. 
The following Table shows the results of tests of a series 
of short wrought -iron beams of circular section made by the 
author when wrought -iron pins were used in bridge con- 
struction, but which illustrate markedly the intensities of 
extreme fibre stress found with short spans. It will be 
observed that the spans were 8 inches and 12 inches only 

CIRCULAR BEAMS OF "BURDEN'S BEST" WROUGHT IRON. 



Kind. 



Diameter. 



Span. 



W. 



Elastic. 



Ultimate. 



K. 



Elastic. Ultimate. 



Turned. 
Turned. 
Turned. 
Turned. 
Rough. 
Rough. 
Turned, 
Turned, 
Rough. 
Rough. 
Turned 
Turned 
Turned 
Turned 



Ins. 



25 

25 

■25 

.25 

.00 

.00 

I .00 

I .00 

I .00 

1 .00 

0.75 

0.75 

0.75 

0.75 



Ins. 
12 

8 
12 

8 
12 

8 
12 

8 
12 



Lbs. 
3,000 
4,400 



1,700 
2,800 

700 
1,200 

700 
1,300 



Lbs. 

6,000 

10,500 



3,000 
4,800 
1,100 
1,900 
1,100 
1,900 



Lbs. 
46,950 
45,900 
54,760 
52,150 
55,000 
57,000 
55,000 

51,950 
57,000 
47,100 
53,880 
47,100 
58,370 



Lbs. 
93,900 

109,500 
93,870 

114,700 
91,700 

101,900 
91,600 

107,000 
91,680 
97,800 
74,050 
85,310 
74,050 
85,310 



558 BENDING OR FLEXURE. [Ch. XII. 

while the diameters of the circular beam sections varied 
from 1.25 inches down to .75 inch. 

W is the centre load and the extreme fibre intensity k 
is computed by eq. (6). The ultimate intensity k was 
assumed to be reached when the deflection at the centre of 
span amounted to about the diameter of the circular, section 
of the beam. This. particular feature of the tests is a matter 
of judgment, but k would differ little whether it be taken 
at a centre deflection equal to the diameter of the circular 
section or one half that diameter or even less. 

It will be noticed that the ultimate values of k are all 
much larger for the 8-inch span than for the 12-inch, and 
that all the ultimate values increase materially with the 
depth of the beam, rising to 107,000 to 114,700 pounds 
per square inch for diameters (i.e., depths of beams) of i 
inch and i{ inch. It will also be observed that the elastic 
limits are greatly increased. The ultimate tensile resist- 
ance of the iron used in these tests was about 55,000 
pounds per square inch and the elastic limit a little more 
than half that value. 

Steel. 

Investigation by actual test has shown that short steel 
beams with circular or rectangular section will exhibit the 
same elevation of ultimate intensity of flbre stress k as 
found for wrought iron in the preceding section. This is 
well illustrated by the following tabular statement of results 
of tests of Bessemer steel beams with circular cross-section, 
also made by the author in the early days of the use of steel 
for bridge building. 

The Table is self-explanatory in view of the explanations 
made for short wrought-iron beams of circular section. The 
ultimate tensile resistance of the mild Bessemer steel used 
in these tests was about 65,000 to 70,000 pounds per square 



Art. 91.] BEAMS WITH RECTANGULAR AND CIRCULAR SECTIONS. 559 
CIRCULAR BESSEMER STEEL BEAMS, EQ. (6). 



Kind. 


Diameter. 


Span 


W 


k. 




Elastic. 


Ultimate. 


Elastic. 


Ultimate. 


Turned 


In. 
1. 00 
1. 00 
1. 00 
I .00 
0.75 
0.75 
0.75 
0.75 


Ins. 
12 

8 
12 

8 
12 

8 
12 

8 


Lbs. 

2,500 
3,750 
1,150 
1,800 
1,150 
1,800 


Lbs. 

4,500 
7,500 
1,800 
3,300 
1,700 
3,300 


Lbs. 
86,000 
85,300 
76,400 
76,400 
77,400 
80,800 
77,400 
80,800 


Lbs. 
146,750 
152,800 
137,520 
152,800 
122 200 


" 


SI 


t( 


148,200 
114,400 
148,200 


(( 


« 





inch and the elastic limit about 35,000 to 38,000 pounds 
per square inch. The ultimate intensity of stress in the 
extreme fibres of these beams ranged, however, from 114,400 
up to 152,800 pounds per square inch, the larger values 
belonging to the greater depth of beam and the smaller 
values to the smaller depth. The elastic limit is seen to 
be correspondingly high. 

These and the preceding tests show that the apparent 
ultimate resistance of wrought iron and structural steel in 
the extreme fibres of very short beams with circular or 
rectangular cross-section may be even more than twice the 
ultimate tensile resistance as derived from the testing of 
ordinary tensile specimens. 

This feature becomes even more marked when the 
spans of the cylindrical beams are still shorter, perhaps 
as short as the diameter of the circular section. 

In the design of pins in pin-connected bridges, this high- 
resisting capacity of wrought iron or steel in pins is recog- 
nized by making the working resistance in the extreme 
fibres of pins considered as beams as much as 50 per cent, 
higher than in members subjected to simple or direct 
tension. 



56o BENDING OR FLEXURE. [Ch. XII. 

The explanation of this phenomenally high resistance to 
the tension of flexure (and also the compression) is found, 
as already indicated, in the fact that the common theory 
of flexure is not correctly applicable to such excessively 
short beams. No such high intensity of tensile (or com- 
pressive) stress actually exists in the metal as computed by 
eqs. (5) and (6). When the span becomes very short, not 
more than perhaps three or four times the depth of the 
beam, lines of stress run from the point of application of 
the load at the centre of the span direct to both supports, 
transverse shear being the vertical components of the 
stresses acting along these lines. All such or similar stress 
action reduces the actual flexure and makes the bending 
stresses of tension and compression correspondingly less; 
but as the flexure formulae, eqs. (5) or (6), contain no 
recognition of this condition, the apparent fibre stresses 
computed by their use are far above the actual. 

Numerous other similar short solid beam tests have 
confirmed the results given in the preceding two Tables. 

Cast Iron. 

Although cast iron is rarely ever used to resist flexure 
except in window and door lintels or other similar members 
whose duties are light, tests of short cast-iron beams have 
shown the same phenomena of greatly elevated ultimate 
resistance as found for the more ductile metals. The 
apparent ultimate intensity k in the extreme fibres of short 
cast-iron beams of circular or square section may be taken 
50 per cent, above the ultimate tensile resistance of the 
same metal under ordinary tensile tests. 

Alloys of Aluminum. 

Table VIII of Art. 59, in the fifth column from the left 
side, exhibits values of the ultimate stress in the extreme 



Art. 91.] BEAMS WITH RECTANGULAR AND CIRCULAR SECTIONS. 561 

fibres of small beams of varying proportions of aluminum- 
zinc alloys. As might be anticipated, beams of either of 
those metals showed comparatively low resistance, but 
with aluminum varying from 80 down to 50 per cent, and 
zinc from 20 up to 50 per cent, the resistance was excel- 
lent, the maximum being found with Al js and zinc 25. 

Table XI of Art. 59 exhibits the ultimate fibre stresses 
in small beams of the alloys of aluminum with copper, zinc, 
manganese and chromium. The rolled bars of Al 96 and 
Cu 4 give excellent results; as does the cast bar of ^/ 75.7, 
Cu 3, zinc 20 and Man 1.3. The remaining values of the 
transverse resistances in the table are self-explanatory. 

Copper, Tin, Zinc, and their Alloys. 

In the following table are given the data and the results 
of the experiments of Prof. R. H. Thurston, as contained in 
his various papers, to which reference has already been 
made. The distance between the points of support was 
twenty -two inches, while the bars were about one inch 
square in section, and of cast metal. 

The modulus of rupture, k, is found by eq. (5), in 
which, however, in many of these cases, W is the weight 
applied at the centre, added to half the weight of the bar. 
When k is large and the specimens small, this correction 
for the weight of the bar is unnecessary ; otherwise it is ad- 
visable to introduce it. 

The coefficient of elasticity, E, is found by eq. (9), in 
which W is the centre load added to five eighths of the 
weight of the bar. 

The manner in which both these corrections arise is com- 
pletely shown in Case 2 of Art. 28. 

E, for any particular bar, has a varying value for dif- 
ferent degrees of stress and strain. Those given in the table 



562 



BENDING OR FLEXURE. 



[Ch. XII. 



SQUARE BARS. 



Percentage of 


*. 


Elastic over 


Final 


E, 








Lbs. per 


Ultimate. 


Deflection. 


Lbs. per 


Cu. 


Sn. 


Zn. 


Sq. In. 


Sq. In. 












Ins. 




lOO 


0.00 


0.00 


29,850 




8.00 


9,000,000 


lOO 


0.00 


0.00 


25,920 


i 0. 14 

] to 0.41 


1.38 
to 8 . 00 


|- 10,830,600 


lOO 


0.00 


0.00 


21,251 


0.346 


2.31 


13,986,600 


lOO 


0.00 


0.00 


29,848 


0. 140 


Bent. 


10,203,200 


90 


10.00 


0.00 


49,400 


0.400 


Bent. 


14,012,135 


90 


10.00 


0.00 


56,375 


0.41 


3.36 




80 


20.00 


0.00 


56,715 


0.657 


0.492 


13,304,200 


70 


30.00 


0.00 


12,076 


I. 00 


0.062 


15,321,740 


61.7 


38.3 


0.00 


2,761 


I .00 


0.032 


9,663,990 


48.0 


52.0 


0.00 


3,600 


I .00 


0.019 


17,039,130 


39-2 


60.8 


0.00 


8,400 


I .00 


0.060 


12,302,350 


28.7 


71.3 


0.00 


8,067 


0.583 


0. 121 


9,982,832 


9-7 


90.3 


0.00 


5,305 


0.25 


Bent. 


7,665,988 


0.00 


100 


0.00 


3,740 


0.273 


Bent. 


6,734,840 


0.00 


100 


0.00 


4,559 


0. 267 


Bent. 


5,635,590 


80.00 


0.00 


20.00 


21,193 




3.27 


11,000,000 


62.50 


0.00 


37.50 


43,216 




3.13 


14,000,000 


58.22 


2.30 


39.48 


95,620 




1.99 


1 1 ,000,000 


55- 00 


0.50 


44.50 


72,308 








92.32 


0.00 


7.68 


21,784 


0.30 


Bent. 


13,842,720 


82.93 


0.00 


16.98 


23,197 


0.41 


Bent. 


14,425,150 


71 .20 


0.00 


28.54 


24,468 


0.51 


Bent. 


14,035,330 


63-44 


0.00 


36.36 


43,216 


0.53 


Bent. 


14,101,300 


58.49 


0.00 


41.10 


63,304 


0.48 


Bent. 


11,850,000 


54.86 


0.00 


44.78 


47,955 


0.39 


Bent. 


10,816,050 


43.36 


0.00 


56.22 


17,691 


I .00 


0.0982 


12,918,210 


36.62 


0.00 


62.78 


4,893 


I .00 


0.0245 


14,121,780 


29. 20 


0.00 


70.17 


16,579 


I. 00 


0.0449 


14,748,170 


20.81 


0.00 


77.63 


22,972 


I .00 


0.1254 


14,469,650 


10.30 


0.00 


88.88 


41,347 


0.73 


0.5456 


12,809,470 


0.00 


0.00 


100.00 


7,539 


0.57 


0.1244 


6,984,644 


70.22 


8.90 


20.68 


50,541 




0.4019 


14,400,000 


56.88 


21.35 


21.39 


2,752 




0.0146 


14,800,000 


45.00 


23.75 


31-25 


6,512 




0.0150 


7,000,000* 


66.25 


23.75 


10.00 


8,344 




0.0162 


12,000,000* 


10.00 


50.00 


40.00 


21,525 




Bent. 


9,000,000 


58.22 


2.30 


39.48 


95,623 




2.000 


10,600,000 


60.00 


10.00 


30.00 


24,700 




0. 1267 


14,506,000 


65.00 


20.00 


15.00 


11,932 




0.0514 


17,000,000 


70.00 


10.00 


20.00 


36,520 




0.1837 


15,000,000 


75.00 


5.00 


20.00 


55,35S 




Bent. 


13,000,000 


80.00 


10.00 


10.00 


67,117 




Bent. 


13,500,000 


55- 00 


5.00 


44 • 50 


72,308 




Bent. 


11,000,000 


60.00 


2.50 


37 • 50 


69,508 




1.500 


13,000,000 


72.52 


7.50 


20.00 


51,839 




Bent. 


12,000,000 


77.50 


I 2 . 50 


10.00 


61,705 




0.705 


13,500,000 


85.00 


12.50 


2.5 


62,405 





B?nt. 


12,500,000 



These bars were about half the length of the others. 



Art. 91.J BEAMS WITH RECTANGULAR AND CIRCULAR SECTIONS. 563 

may be considered average values within the elastic 
limit. 

As usual, "elastic over ultimate^' is the ratio of k at the 
elastic limit over its ultimate value. 

An examination of the ultimate tensile and compressive 
resistances of these same alloys, as given in preceding pages, 
shows that the. ratio of k over either of those resistances is 
very variable. It is usually found between them, but occa- 
sionally it exceeds both. 

Timber Beams. 

As timber beams are always rectangular in section, eq, 
(3) only will be needed. Retaining the notation of that 
equation, if the beam carries a single weight W at the centre 
of the span /, 

^.. 2 kAh , ^ 

^ = 3— ^'^) 

If the total load W^ is uniformly distributed over the 
span, 

1^'=^^ (II) 

As k is supposed to be expressed in pounds per square 
inch, all dimensions in eqs. (10) and (11) must be expressed 
in' inches. 

In the use of timber beams it is usually convenient to 
take the span / in feet, and the breadth (b) and depth 
(h) in inches. Placing 12/ for /, therefore, in eqs. (10) 
and (11), 

^^, kAh J ,_, kAh . . 

W=^; and W =2^ (i.) 



564 BENDING OR FLEXURE. [Ch. XII. 

in which formulae / must be taken in feet and A and h in 
inches. 

k 

If B be put for — eq. 12 becomes 
-^ 18' 

W = B^; and W = 2B^. . . . (13) 

Hence when W and W have been determined by ex- 
periment, 

For single load W at centre 

^ Wl , Wl 18I/F/ \Wl \Wl 

^ = Ah •'• ^=AB=-Ak~=ym=^-'^^^' (^4) 

For total load W uniformly distributed 

Wl Wl gW'l \Wl \ Wl 

^~2Ah ''' '^~2AB~ Ak '~\2Bh~^S kh' ^'5) 

If the beam has a section one inch square and is one foot 

W 
long, B = W = — . B, therefore, may be considered the 

unit of transverse rupture ; it is sometimes called the coefficient 
for centre-breaking loads. 

If the depth h of the beam is given and the breadth is 
desired, eq. (14) gives 



Wl 18M 
b = 

Eq. 15 also gives 



^ = gF=^^- • • • • • (16) 



, Wl gW'l , , 

^=^5F=-W • • • • • (^7) 

In general, whatever m.ay be the distribution of the load- 
ing, if the bending movement is M (in inch-poimds), eq, 
(3) gives 



Art. 91.] BEAMS WITH RECTANGULAR AND CIRCULAR SECTIONS. 565 



^--iBh 



or 



M 6M , . 

^=^5P^W ^^9) 

The general observations which have already been made 
in connection with the ultimate resistances of timber in 
tension and compression are equally applicable to the flex- 
ure or bending of timber beams. The ultimate resistance 
of the timber as exhibited by the intensity of stress in the 
extreme fibre can safely be taken only when determined 
from tests of full-size beams as actually used in engineering 
structures. Such resistances or moduli when determined 
from small pieces selected for the purpose of test are liable 
to be largely in error for the reasons given in detail in Art. 6i. 
In fact Messrs. Cline and Heim state in Bulletin io8, 
" Tests of Structural Timbers," U. S. Department of Agri- 
culture, that values obtained from testing small thoroughly 
seasoned selected specimens " may be from one and one 
half to two times as high as stresses developed in large 
timbers and joists," and that statement is rather under than 
over, as many tests have shown. Furthermore, it is essen- 
tial to know at least approximately the degree of seasoning 
to which the timber has been subjected. Ordinary air 
seasoning will seldom reduce the moisture in full-size timber 
beams to less than 15 per cent, to 20 per cent. Inasmuch 
as timber in open engineering structures, like bridges, will 
at all times be exposed to rainfalls often heavy, working 
stresses used in the design of such structures should be 
prescribed for wet or green condition. If the structure is 
to be protected from atmospheric moisture, values belong- 
ing to seasoned timber may properly be employed. 

Table II of Art. 61 gives the modulus of rupture for 



S66 



BENDING OR FLEXURE. 



[Ch. XII. 



full-size beams tested to failure on a span of 1 5 feet by con- 
centrated loading at two points one third of the span from 
each end (Messrs. Cline and Heim, U. S. Dept. Agri- 
culture). These results include failures by tension and 
compression of fibres as well as failures due to shear along 
the neutral surface of the beams. Both green and air- 
seasoned timbers were tested with the sections given in the 
Article cited. 

Table I gives the results of the same series of tests under 
a proposed grading by which all beams tested were divided 
into Grade I and Grade II, the higher resistances being 
found in the former. 

Table I. 

AVERAGE RESISTANCE VALUES OF DIFFERENT SPECIES BY 
PROPOSED GRADES 











Average 
Modulus of 


Average 














Fibre Stress 


Average Modulus 




Nam 


ber of Tests. 


Rupture per 
Square Inch. 


at Elastic 


of Elasticity per 


Species. 








Limit per 
Square Inch. 


Square 


Inch. 




Total. 


Grade 


Grade 


Grade 


Grade 


Grade 


Grade 


Grade 


Grade 






L 


n. 


L 


n. 


I. 


II. 


I. 


II. 










Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Longleaf pine . . 


17 


17 




6,140 




3.734 




1,463.000 




Douglas fir. . . . 


161 


81 


80 


6,919 


5,564 


4.402 


3,831 


1,643,000 


1,468,000 


Shortleaf pine. . 


48 


35 


13 


5.849 


4.739 


3.318 


3.005 


1,525.000 


1,324,000 


Western larch. . 


62 


45 


17 


5,479 


3,543 


3.662- 


2.432 


1,365,000 


1,130,000 


Loblolly pine . . 


94 


45 


49 


5,898 


4.702 


3.513 


2,793 


1,535.000 


1,309,000 


Tamarack 


25 


9 


16 


5,469 


4.52s 


3. 151 


2,847 


1,276,000 


1,261,000 


West, hemlock. 


39 


26 


13 


5,615 


4.658 


3.689 


3.172 


1,481,000 


1,360,000 


Redwood 


28 


21 


7 


4,932 


3.091 


4,031 


2,947 


1,097.000 


877,000 


Norway pine. . . 


34 


17 


17 


4,821 


3.764 


3.082 


2,364 


1.373.000 


1,204,000 



The intensities of stresses in extreme fibres are averages 
for each kind of timber at rupture and at elastic limit. It 
is to be understood, however, that the elastic limit is approxi- 
mate only as it is not a well-defined point in timber. 
The moduli of elasticity are fully as high as should be taken, 
if, indeed, they are not a little too high for ordinary pur- 
poses. 



Art. 91.] BEAMS WITH RECTANGULAR AND CIRCULAR SECTIONS. 567 

The table does not include results for white pine and 
spruce, but the resisting and elastic qualities of those two 
timbers are so near to the corresponding qualities of Norway 
pine that, they may be assumed to be the same under ordi- 
nary conditions. 

Table II gives a summary of the results of tests of full- 
size beams made by Prof. Arthur N. Talbot and described 
by him in Bulletin No. 41 (1909) of the University of Illinois. 
The cross-sections of these beams varied from 7 inches by 
12 inches to 8 inches by 16 inches and the spans were 13.5 
feet and 14.5 feet. The loads were applied equally at two 
points, each one third of the span from each end. 

The series into which the program of results is divided 
were used as a matter of convenience only and have no sig- 
nificance as to quality of material or as to physical features 
of the results. 

It will be observed that small beams and shear blocks 
were also tested and that the results for these smaller pieces 
are on the whole materially larger than for the full-size 
beams and nearly or quite twice as large in some cases. 

The extreme fibre stress was computed by means of 
eq. (5), in which VV is the total load at the two points of 
application at failure and / is two-thirds of the actual length 
of span in the tests, which makes the bending moment 
M =^WL If this external bending moment is placed equal 

2kl 

to the — — , the intensity of stress k will take the value, as 

h 

indicated by eq. (5) : 

In this equation h is the depth of the beam and b its 
breadth, as already explained in connection with eqs. (i) 
and (la). W is obviously the load given by the reading 



S68 



BENDING OR FLEXURE. 



[Ch. XII. 



c/3 . 
O ^ 






O Ov M ir> 1/0 Tt 

rO C^ lO 



fO M lO O w O 
(^ O t^ -^ lO Ov 

ro ^ M 0_ t-- O 



r0U000OOOO>J1>O 
0,1000 lOOOO r-'^i-i 
tNroOMi-i^O\'t<r> 



o g£ 
Q 



QO^ 



'u O 



oV. 



<U 1) 









00 ro w O ro O 
Oi 00 M ~^ O O 
01 ro M l^ ■^ ro 



mojOOOOvdOvO 
"100 OivO <N t^-^OOO 



M lO VO O t~- 00 

M 0) t^ (T) 00 •^ 

rr) ir> t-~ 



ro w O O ro O 
n ro (~0 O M Tf 



O^OOOOr0>OO^-| 
MNOvOOOOO-^W'* 

ro Tt o <3 q fo q> o)_ ■* 



■* O O Ov o o 

M lAJ lA) 00 \0 O 

M On O 



1^ CO l> 
lO rf fO 



rOvO ^ O O »Oir>>on 



00 r^ O vO 
ro lO "^ 't 



(^ 00 "* vo r- O 
r^ \0 ts " 00 M 
M n <N O O O 



O fO rO O lO 
N -^ ■<* t^ t^ 

P4 0) O 00 M 



IT) lO 00 -^O 

C^ CN \0 O f^ 

ro w O 



O 00 CO in N O 
r^ 00 vO O 00 O 
ro 1-0 ro " O 00 



't -+ o o o 

rO ro O O O 
ro ro m lO lO 



lO lO IN \0 00 

t^ M O 00 r^ 

ro ro O 



O ■* O M ID 
M OJ lO 00 M 
■* (N in O O 



M r^ i^ in O 

o mo O ro 
ro fj O ^ 1^ 



O O "* r^O 
O vO '^OO 
w fO fO 00 



■* in ro o O in 

sD O O i-i 00 O 
ro in <N 00_ 00 0_ 



'd' rO t^ O M O 

M Os 00 O t- 

(-0 00 •^ 



O ^ t- O ro O 
O l> ro ^ 00 O) 
ro -rl- 01 in '^ O 



Tf ^ O O O 00 O 
^ 't O O O w w 
00 oo 00 ro ro m M 



000-^000 Ol^oOOt^O 

oivO-^^OO t^OoOOM-* 

oooooo ro^MdinvO 

rf in t^ in oj 



sOwoOOOOinOO 

o t^t^o *-" inooMvo 
oO'^Ooo •^inoooo •* 






^ ei^ 



^^ 5 






03 e 



(U 



':2 C rrt oj •« 



i2 g.2 

130 fc C 
'to O O 



ceo 

CO to -r-; 



tw (1) (u O ^ XI 



U o 
« l-i i-< 
J (U 0) 



u, ,^ u, -^ 

tiO be M M 

oj n! nS c 

fu m aj rS 

> > > CTl 



g§1 



> •" O 



^ >iS "rt 

-M to 

a! CO CO 

<U to 

1-, d) 



E 2i 



•5 ^ 

c3 :^ 



CO 



— CO to 

nJ 1) to 

^ ^^ OJ 

C -►-> I- 



'C JH *=3 

"5 <U ^ to 

5i ^ +j to 

to ™ 1-1 

rt I ?5 "^ 

c o 1;5 "^.S 

^ .2 0) « !3 

**- x; cc y= to 

"ti OJ Qj OJ aj 

'^ tx S) M bo 

!0 ni cti nl c3 

^ u XL ^ u 

0) (U (U K D 

i > > > > 



S « & 

bo<; hJ 



be 






bo ^ ^ M^ ^ :=: I- 
ffi^^ffi g g g « 

C/2 W W W 



Art. 91.] BEAMS WITH RECTANGULAR AND CIRCULAR SECTIONS. 569 

of the scale beam of the testing machine. If Wi is one of 
the two equal loads applied to the beam at each one third 
point of the span, 2W1 must be written for W. 

The ultimate intensity of shear shown in Table II, which 
is both the intensity of shear in the neutral surface and on 
a normal section of the beam at the same point, is found by 
simply taking one and one half the end reaction divided 
by the cross-section bh of the beam. As the total transverse 
shear is greatest at the end of the span, the greatest inten- 
sity of shear on the neutral surface will be found at that 
point at or near which failure by shear will begin unless 
induced elsewhere by a season crack, wind-shake, decay 
or some other weakness of the material. Obviously there 
is neither transverse nor longitudinal shear between the 
two points, equally loaded, as they are symmetrically located 
with reference to the centre of the span. 

Table III shows the moduli of elasticity computed by 
Professor Talbot from the data secured by his beam tests. 
The modulus is found by observing the centre deflection of 
the beam when loaded within its elastic limit and then 
inserting the observed value of the deflection and the cor- 
responding observed load in a formula similar to eq. (7). 
Eq. (7) itself is not applicable for the reason that these 

Table III. 



Timber. 


Modulus of Elasticity (£), 


Max. 


Mean. 


Min. 


Longleaf pine. 


2,105,000 
1,595,000 
1,478,000 
1,915,000 
1,857,000 
2,087,000 
1,900,000 


1,620,000 
1,591,000 
1,229,000 
1,386,000 
1,251,000 
1,780,000 

1,499,000 


1,025,000 

1,585,000 

887,000 

944,000 

611,000 


Shortleaf pine, untreated 

Shortleaf pine, creosoted 

Loblolly pine, untreated 


Loblolly pine, creosoted 


OIH Dourias fir 


1,310,000 
1,138,000 


New Douglas fir 





S76 



BENDING OR FLEXURE. 



tCh. XII. 



beams were not loaded at the centre of span. The formula 
for the centre deflection, however, is readily derived by 
an analysis similar to that used in Art. 28. That operation 
will give 

23WP . ^ 23M3 



w = 



i2g6EI 



or 



I2g6lw' 



The preceding experimental values for timber are among 
the latest determinations and are representative of the 
best engineering practice, especially as they are based on 
tests of full-size timbers of as good quality as can probably 
be secured in the open market. 

The American Railway Engineering Association, after 
careful scrutiny of all tests of timber made up to 191 1, 
recommended the values given in Table IV for use in the 



Table IV. 
UNIT STRESSES IN POUNDS PER SQUARE INCH 





Bending. 


Shearing. 


Timber. 


Extreme Fiber 
Stress. 


Modulus 

of 
Elasticity. 

Mean. 


Parallel to 
the Grain. 


Longitudinal 
Shear in 
Beams. 




Mean 

Ult. 


Working 
Stress. 


Mean 
Ult. 


Working 

Stress. 


Mean 
Ult. 


Working 
Stress. 


Douglas fir 

Longleaf pine .... 
Shortleaf pine. . . . 

White pine 

Spruce 

Norway pine 

Tamarack 

Western hemlock . 

Redwood 

Bald cypress .... 

Red cedar 

White oak 


6,100 
6,500 
5.600 
4.400 
4,800 
4,200 
4,600 
5,800 
5,000 
4,800 
4,200 
5.700 


1,200 
1,300 
1,100 
900 
1,000 

800 
900 

1,100 

900 
900 
800 

1,100 


1,510,000 
1,610,000 
1,480,000 
1,130,000 
1,310,000 
1,190,000^ 
1,220,000 
1,480,000 

800,000 
1,150,000 

800,000 
1,150,000 


690 

720 

710 

400 

600 

590* 

670 

630 

300 

500 

840 


170 
180 
170 
100 
150 
130 
170 
160 
80 
120 

210 


270 
300 
330 
180 
170 
250 
260 
270* 

270 


IIO 
120 
130 
70 
70 
100 
100 

100 

IIO 



Unit stresses are for green timber and are to be used without increasing 
the live load stresses for impact. Values noted * are for partially air-dry 
timbers. 



Art. 91.] BEAMS WITH RECTANGULAR AND CIRCULAR SECTIONS. 571 

design and construction of timber railway structures for 
the modulus of elasticity in flexure, the ultimate resistance 
and working stress in extreme fibres of bent beams, and 
similar quantities for ordinary shearing parallel to the grain 
and for longitudinal shearing along the fibres in the neutral 
surface of beams. 

The intensities of working stresses given in this Table 
are for railway structures. It may be justifiable to use 
somewhat higher values in other structures where the mov- 
ing loads are more steady or where perhaps it may be proper 
to consider all loading as practically quiescent or dead load. 
It is always to be remembered, however, that timber struc- 
tures are usually highly combustible and hence that it 
will frequently be advisable to provide some surplus of 
sectional area to prolong the carrying capacity of timber 
members after the beginning of a fire. 



Failure of Timber Beams by Shearing Along the Neutral 

Surface. 

In the preceding treatment of timber beams, it has been 
assumed that when broken under test the extreme fibres 
will fail, either in tension or compression. As a matter of 
fact, failure of such beams usually takes place at some weak 
spot, as a knot, point of incipient or active decay, or at some 
other point where abnormal weakness is developed. This 
latter observation holds true whether the. failure of the beam 
takes place by tension or compression in the extreme fibres 
or by shearing in the neutral surface. 

In Art. 15 it was shown that the greatest intensity of 
either transverse or longitudinal shear in any normal sec- 
tion of a beam takes place at the neutral surface, and hence 
that the tendency of the fibres there is to separate by longi- 



572 BENDING OR FLEXURE, [Ch. XII. 

tudinal movement over each other. This is precisely the 
kind of failure which actually takes place in some short tim- 
ber beams. If the total transverse shear at any normal sec- 
tion of the beam is 5, eq. (8) of Art. 15 shows that the 
maximum intensity, s, of shear in the neutral surface is 

' = ijr^- • (20) 



bd 



In this equation, b is the breadth or width of the beam 
and d the depth, usually taken in inches. 

If W is a single weight or load at the centre of span of a 
beam simply supported at each end, the shear s, as far as 
that single load is concerned, is constant throughout the 
entire length of the beam with the value 

3W 

If, again, the beam is uniformly loaded with the total 
load W\ the intensity of shear 5 in the neutral surface has 
a value which varies from zero at the centre of span to the 
value given by eq. (21) after making W = VV\ Whenever 
the value of the intensity s exceeds the ultimate intensity 
of shear along the fibres lying in the neutral surface, the 
beam will fail by the separation of its two halves or parts 
at the neutral surface. 

The mean values for the ultimate resistance to shear 
along the fibres in the neutral surface of his loaded beams 
were found by Prof. Talbot and are given in Table II for 
the best varieties of pine timber and for Douglas fir, in- 
cluding results for creosoted beams of shortleaf pine and 
loblolly pine. The values for shear and other quantities 
recommended by the American Railway Engineering Associ- 
ation are found in Table IV. 



Art. 91. 



SHEARING ALONG NEUTRAL SURFACE. 



573 



The average values of the ultimate shear in the neutral 
surface determined by Messrs. Cline and Heim in their 
" Tests of Structural Timbers," already cited, are given in 
Table V for nine varieties of structural timbers, both green 
and air-seasoned. These results belong to the same full- 
size beams as the values given in Table I of this Article. 



Table V. 

COMPUTED SHEARING STRESSES DEVELOPED IN STRUCTURAL 

BEAMS 





Total 
Number 
of Tests. 


First Failure by Shear. 

Per cent, of Total and 

Average per Sq. In. 


Shear Following Other 

Failure. 
Per cent, of Total and 
Average per Sq. In. 


Species. 


Green. 


Dry. 


Green. 


Dry. 


Green. 


Dry. 




% 


Lbs. 


% 


Lbs. 


% 


Lbs. 


% 


Lbs. 


Longleaf pine 

Douglas fir 

Shortleaf pine 

V/estern larch 

Loblolly pine 

Tamarack . . . 


17 
191 

48 

62 
III 

30 

39 

28 

49 


9 
91 
13 

52 

25 

9 
44 
12 
10 


54 

2 

17 

8 

7 
10 

5 
7 
6 


353 
166 

332 

288 

335 
261 
288 
302 
232 


56 

6 
46 

27 
28 

33 

23 



10 


272 
221 
364 
340 

434 
299 

307 

278 


23 
22 

6 

16 

2 

28 

II 

6 


374 
295 
327 
314 
356 
263 
281 
218 
266 




49 
8 

21 

16 


68 

17 



294 
418 
370 
546 


Western hemlock . . . 
Redwood 


438 
250 


Norway pine 



It will be observed in all of these tests that there is 
much variation in the intensities of the different stresses 
found and especially in these ultimate intensities of shear 
in the neutral surfaces of full-size beams. As has already 
been indicated this is due to the presence of a variety of 
weakening defects to which timber is subject. This sig- 
nifies that low working stresses should be used. 

It has been found in many cases, and possibly in nearly 
all, that wind-shakes, season cracks, and other influences 



574 BENDING OR FLEXURE. [Ch. XII. 

which induce at least partial separation of the fibres at the 
neutral surface, are the sources of incipient failure by shear- 
ing in the neutral surface. 

In designing timber beams this liability to shear along 
the neutral surface should always be carefully tested by 
computations. Relatively short beams are particularly 
liable to fail in this manner, and the greater part of the 
timber beams used in engineering work are of this 
class. 

It is a very simple analytical matter to establish such 
a relation between the methods of failure by longitudinal 
shearing and rupture of the fibres as to indicate more or 
less approximately the limit beyond which one mode of 
failure is more liable to occur than the other, but empirical 
values for both these ultimate resistances have been seen 
to be so variable as to make it more advisable to compute 
the carrying capacity of the beam by both methods, especi- 
ally as each is a simple procedure. 



Influence of Time on the Strains of Timber Beams. 

It has been found by actual observation that if a timber 
beam is loaded to no greater extent than one fourth of its 
ultimate load, the resulting deflection will continue to in- 
crease under continued loading for a long period of time. 
Sufficient investigations have not yet been made to express 
these results quantitatively Avith much accuracy. Enough 
has been ascertained, however, to show that the influence 
of time is most important in determining the deflection of 
timber beams under loads applied for a considerable period 
of time, and that when the loading becomes a large portion 
of the ultimate, i.e., perhaps 75 per cent., the beam may 
fail if the application be sufficiently continued. Indeed, 



Art. 91.] CONCRETE BEAMS. 575 

some experiments indicate that failure may possibly take 
place at .6 or .7 of the ultimate of a single application, if 
that amount be imposed a sufficient length of time. 

It should be understood, therefore, that in using the co- 
efficients of elasticity given in this article for the purpose 
of computing deflections, such computations may be applic- 
able only when the loads are applied for short periods of 
time. 

Concrete Beams. 

When a concrete or a natural stone beam is subjected to 
transverse loading it fails by tearing apart on the tension 
side. The failure of the beams, therefore, indicates to some 
extent the ultimate tensile resistance of the material. Ob- 
viously, in the case of concrete beams the ultimate carrying 
capacity will depend upon a number of elements, such as 
the kind and quality of cement, sand and broken stone used, 
and the proportions of the mixture. Table VI contains 
results of tests of a considerable number of concrete beams 
6 ins. by 6 ins. in cross-section and six months of age. For 
three months these beams were frequently wetted though 
kept in air. During the remaining three months they were 
kept in air without wetting. The length of span for some 
of these beams was 42 ins. and 18 ins. for the remainder. 
Within the limits of the tests this difference in span appeared 
to make no essential difference in the ultimate intensities 
of stress in the extreme fibres. With the cross-sections of 
the beams, i.e., 6 ins. wide and 6 ins. deep, the ratio of span 
length divided by the depth was either 7 or 3, making the 
beams very short. The different columns of the table show 
the character of the ingredients of the concrete as well as 
the greatest, mean, and least values of the intensities of ex- 
treme fibre stress K. As would be anticipated, the values 



576 



BENDING OR FLEXURE. 



[Ch. XII. 



Table VI. 

CONCRETE BEAMS vSIX MONTHS OLD. 





Concrete. 


Size of 
Stone in 
Inches. 


No. of 
Tests. 


Ultimate Stress in 
Extreme Fibres. 
Lts. per Sq. In. 




Max. 


Mean. 


Mm. 


B'klyn Bridge Rosendale 

(1 ( ( < ( 

Atlas Portland 


c. s. br. 
.1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

1-3-5 

1-2-4 

T-,^-5 


0-2^ 

1-2^ 

O-I 

i < 

0-2j 

1-2* 

O-I 

0-2^ 

172^ 

O-I 

0-2^ 

1-2^ 
O-I 

Gravel. 


6 
5 
4 
3 
6 

3 
6 
6 
6 
6 
6 
6 

5 
6 
6 
6 
6 

5 
6 
6 
6 
6 
6 
6 
6 
I 
6 
6 
6 
6 

5 
6 


140 
128 

153 
140 

134 
].« 

647 
516 
510 

458 
560 
516 
385 
329 
554 
297 

423 
329 
560 
491 

574 
460 

654 
541 
192 

554 
417 
379 
279 
460 

^73 


103 
80 
128 
136 
125 
126 
526 
449 
452 
402 

503 
420 

349 
283 
424 
268 

377 
272 
472 
404 

493 
419 
566 

484 
171 

157 
481 

352 
344 
245 
382 
'^14 


76 

33 
109 

134 
120 
122 
460 
360 

335 
360 

458 
355 
282 


1 ( 1 ( 


< ( < 1 


( ( < ( 


< ( < ( 


.. 


Silica Portland 




238 
326 
224 
297 
238 
404 

341 
453 
391 
466 
414 
147 

414 
285 
312 
195 
326 
266 


t < i< 


( ( < ( 


<( (( 


.< 


Alsen Portland . 


( ( ( ( 


(< << 


(1 <i 


.. 


«< <c 


B'klyn Bridge Rosendale 

Atlas Portland 




Silica Portland 




Alsen Portland 













" c. " indicates cement; "s." indicates sand; "br." indicates broken stone 
or gravel. An excellent limestone was used for broken stone. 

for the Portland cement beams are much higher than those 
for the Rosendale cement. The table exhibits the usual 
variations in the results for such material, but on the whole 
those for gravel are seen to be somewhat less than those for 
broken stone, the proportions of mixture being the same 



Art. 91. 



CONCRETE BEAMS. 



577 



for the two materials. Even with Portland cement, and 
with as rich a mixture as 1-2-4, the results show that 
working values of the greatest intensity in extreme fibres 
should not exceed 40 to 60 pounds per square inch. 

The investigations from which the results in Table VI 
have been taken were conducted by Messrs. George C. 
Saunders and Herbert D. Brown, graduating students in the 
class of Civil Engineering of Coltimbia University in 1898. 

The results of tests of twelve Giant Portland cement 
concrete beams with 30- and 6 8 -inch spans are given in 
the " U. S. Report of Tests of :\Ietals and Other Materials " 
for 1900, and they are shown in Table VII. 

Table VII. 

TRANSVERSE TESTS OF GIANT PORTLAND-CEMENT 

CONCRETE BEAMS. 

Composition: i c, 3 s., 5 br. st. 



span. 

Inches. 


Breadth, 

Inches. 


Depth, 
Inches. 


No. of 
Tests. 


Ultimate Stress, k, In Extreme 
Fibres, Pounds per Square Inch. 


Max. 


Mean. 


Min. 


68 
30 
30 


6 
6 and 4 
6 and 4 


6 
6 
6 


I 
7 
4 


564 

454 


472 
493 
415 


348 
367 



The age of these beams was made up of 2 days in air, 
2 months in water, and then i one month in air, making a 
total of 3 months and 2 days. The broken stone included 
all sizes passing a 2^-inch ring, and retained on a sieve with 
i-inch meshes. In these tests the ratio of length over depth 
was 5 , except in the first where it was 1 1 . There seems to 
be little difference in the values of k for the two ratios, but 
the number is much too small to yield any law of variation. 

The values of the ultimate extreme fibre stresses fe, 
shown in Table VIII, are the results of testing to failure 
short Portland cement concrete beams by Mr. H. Von Schon, 



578 



BENDING OR FLEXURE. 



[Ch. x:i. 



Chief Engineer of the Michigan Lake Superior Power Com- 
pany, at Sault Ste. Marie, Mich., and they are taken from 
his paper in the * ' Transactions of the American Society of 
Civil Engineers" for December 1899. The beams were 6 
inches by 6 inches in cross-section, with a span of 18 inches. 
The ratio of length over depth, therefore, was 3. 

Table VIII. 

PORTLAND-CEMENT CONCRETE BEAMS, 6 INS. BY 6 INS. SECTION, 

18 INS. SPAN. 











Ultimate Fibre Stress, K, 










Poimds per Square Inch. 


Cement. 


Broken Stone. 


Mixture. 


No. of 
Tests. 






Max. 


Mean. 


Min. 


E 


Sandstone 


A 


2 


178 


176 


174 




' ' 


B 


2 


225 


217 


209 




( < 


C 


2 


288 


280 


272 




" 


D 


2 


329 


325 


321 




' ' 


E 


2 


108 


102 


97 




Boulder stone 


■ A 


2 


354 


326 


298 




' ' 


B 


2 


358 


328 


299 




( ( 


C 


2 


390 


373 


356 




' * 


D 


2 


420 


410 


400 




' ' 


E 


2 


350 


330 


310 




Sandstone 


A 


2 


181 


169 


158 




( ( 


B 


2 


183 


175 


167 




< ( 


C 


2 


266 


262 


258 




" 


D 


2 


328 


308 


288 




' ' 


E 


2 


195 


182 


169 




Boulder stone 


A 


2 


390 


347 


204 




' ' 


B 


2 


423 


406 


390 




" 


C 


2 


410 


392 


374 




t < 


D 


2 


411 


393 


375 




( ( 


E 


2 


332 


322 


312 



Mixture A . 
B. 
C. 
D. 



cement, 2.4 sand, 5.3 broken stone. 
2.4 " 4.8 " 
" 2.4 " 4-4 " 
" 2.4 " 4 



" E. . . .1 " 0.3 lime, 3.1 sand, 5.3 broken stone. 

The beams were left from two to eight days in their 
forms or moulds after being made, and then tested at the 
age of 60 days in air. 



Art. 91.] CONCRETE BEAMS. 579 

The chief elements in the composition of the Portland 
cements indicated by E and R in the Table were as follows : 

Cement E. Cement R. 

Lime 62 . 38 63 . 55 

Silica 23 . 08 2 1 . 70 

Alumina 5 . 69 8.76 

Magnesia i . 2 1 2 . 96 

Iron oxide 5 . 35 1.27 

Potash and soda i . 66 i . 12 

The sand used in Mr. Von Schon's tests was from St. 
Mary's River, the broken sandstone was the native Pots- 
dam variety, while the broken boulder stone was granitic 
in character. All broken stone would pass through a ij- 
inch ring and be retained on a i-inch ring; the material 
was, therefore, little balanced. 

In the constructions executed under the supervision of 
the Boston Transit Commission, large amounts of concrete 
were needed, and in the report of the Commission for the 
year ending June 30, 1902, there are exhibited a large num- 
ber of tests of Portland-cement concrete beams 6 inches 
by 6 inches in cross-section with 30-inch spans. The ratio 
of length of span over depth of beam in this case is, there- 
fore, 5. Table IX gives the greatest, average, and least 
results of these tests with the number of beams broken. 

Table IX. 

PORTLAND-CEMENT CONCRETE BEAMS, 6 INS. BY 6 INS. SECTION, 

30 INS. SPAN. 





Composition by 








Ultimate Fibre Stress, k. 




Volume. 


Hours in 


Air Pressure, 


No of 


Lbs. per Sq. In. 






Compressed 


Lbs. per 


Tests 
























Air. 


Sq. In. 










Cement. 


Stone 
Dust.* 


Broken 
Stone.* 








Max. 


Mean. 


Min. 


I 




1-7 


2.75 


24 


7-12 


12 


999 


851 


677 


I 




1-9 


2.6 


24 


12-18 


50 


924 


850 


590 


I 




2 


2.4 


48 


18-25 


30 


904 


731 


622 


I 




2 


2.4 


28-30 days 


20-25 


100 


900 


728 


523 



* Approximate volumes. 



58o 



BENDINC OR FLEXURE. 



[Ch. XII. 



The concrete was machine mixed and Vulcanite-Port- 
land cement was used. The stone dust, to which reference 
is made in the table, was finely crushed stone varying from 
impalpable powder up to J inch diameter, the broken stone, 
on the other hand, being of ordinary size. It will be noticed 
that these beams were kept a part of the time in compressed 
air at pressures varying from 7 to 25 pounds, presumably 
for the reason that some of the material was to be used under 
such conditions. 

Table X contains results of a number of tests of con- 
crete beams 6 inches by 6 inches in cross-section and with 
30-inch spans, made for the purpose of comparing the re- 
sistances of concretes made with stone dust and sand. 
This table is also taken from the Report of the Boston 
Transit Commission for the year ending June 30, 1902. 



Table X. 

PORTLAND-CEMENT CONCRETE BEAMS, 6 INS. BY 6 INS. SECTION, 

30 INS. SPAN. 



Composition by Volume (Approximate). 


No. of 


Ultimate Fibre Stress, k. 
Pounds per Square Inch. 










Tests. 








Cement. 


Sand. 


Stone 
Dust. 


Broken 
Stone. 




Max. 


Mean. 


Min. 


I 


— 


2 


2.4 


4 


947 


848 


760 


I 


•9 


•9 


2.7 


4 


846 


784 


704 


I 


1.6 





3 


4 


773 


711 


656 


I 


• 9 


■9 


2.7 


4 


862 


806 


759 



This concrete was also made with Vulcanite-Portland 
cement and the mixing was done by hand. The beams 
were kept in air for the first 24 hours and then 29 days in 
damp earth. 

The results both as to coefBcient of elasticity and ex- 
treme fibre stress, given in Table XI, were determined 
at the mechanical laboratory of the Department of Civil 



Art. 91. 



CONCRETE BEAMS. 



S8i 



Engineering of Columbia University in 1902 by Mr. Myron S. 
Falk.* They have special value from the age of the beams, 
which was about seven years. These beams were originally 
made under the supervision of Mr. A. Black, Instructor in 
Civil Engineering, Columbia University, for the purpose of 
determining thermal linear expansion. They were kept 
well moistened for several months after being made, but 
subsequently until tested they were kept under cover with- 
out moistening. The gravel used was rounded, varying in 
size from ^ to 2^ inches. 



Table XL 

PORTLAND-CEMENT MORTAR AND CONCRETE BEAMS BROKEN 
BY CENTRE WEIGHT. 









Section of Bar 






Bar. 


Age, 
Years. 


Span in 
Inches. 


in Inches. 


Coefficient 

of Elasticity, 

Pounds per Sq. In. 


Extreme 
Fibre Stress, 
Pounds per 
Square Inch 




Depth. 


Width. 


A 


7-4 


36 


4... 


4.06 






^^ 


7-4 


16 






1,591,000 


278 


i' 


7-4 


16 






1,102,000 


315 


B 


7 


36 




4.00 


2,122,000 


606 


Bx 


7 


16 






2,440,000 


636 


?^ 


7 


16 






1,220,000 


530 


C 


7 


36 




405 


1,315,000 


247 


c. 


7 


16 






387,000 


229 


c. 


7 


16 






1,023,000 


208 


D 


7-3 


36 


4. 10 


4-15 


1,165,000 


294 


- D, 


7-3 


16 






597,000 


415 


D2 


7-3 


16 






597,000 


346 



Bars^, I Aalborg cement, 2 sand, 4 gravel. 

" B, I Atlas " 3 " 

■ " C, I Alsen " 3 " 5 gravel. 

" D, I " " 2 " 



Some of the coefficients of elasticity are abnormally 
low, those belonging to the beams 5, B^, and B2 are fairly 



*See Proc. Am. See. C. E., February, 1903. 



582 BENDING OR FLEXURE. ]Ch. XII. 

representative of what may be expected with such mate- 
rial in flexure. 

Plate A represents graphically the results of the tests 
of the preceding three bars B. As usual, the strain of 
deflection consisted of two parts in all cases, one perma- 
nent, at least for the time being, and one elastic, which 
disappeared on the removal of the load. This feature is 
shown by two lines, in each case indicated by the same 
letter and subscript. The difference between the total 
and permanent strain or deflection varied very nearly as 
the centre load, and that dilYerence being the elastic de- 
flection was used in computing the coefficients of elasticity 
given in Table XI. No coefficient of elasticity was com- 
puted for a centre loading less than about 200 pounds 
For the purpose of computing deflections under ordinary 
working stresses from a condition of little or no loading, 
it would be best to take the cceffcient of elasticity at not 
more than one half of the values given in the Table, in 
order to allow for that part of the deflection which does 
not disappear immediately upon the removal of the loading. 

Reviewing all the preceding values of the ultimate 
stress in the extreme fibres of concrete and mortar beams, 
the working intensities of stress in extreme fibres can prob- 
ably not be properly taken higher than 50 "to 75 pounds 
per square inch when Portland cement is used for well- 
balanced mixtures not less rich than i cement, 2 sand, and 
4 broken stone, or possibly, where exceptionally well made, 
I cement, 3 sand, and 5 broken stone. If gravel is em- 
ployed, some reduction should be made, depending upon 
its character, and a similar observation must be applied 
to mixtures less rich in cement than the preceding. 

For natural cements, values of working stress greater 
than one fourth of the preceding probably should not be 
used. Indeed, it may be a serious question whether 



Art. 91. 



CONCRETE BEAMS. 
PI.ATE A. 



583 



































































































































































































































































































150 


3 




. 




















/ 




































/ 






1 






/ 








•7 














140 


) 












/ 




1 




j 










■/ 




























/ 






1 




1 










/ 














130 


) 












! 






1 




1 










/ 




























i 






i 




1 








) 
















120( 


) 










^1 






I 


1 










1 




























' 






i 


f 








A 


iy 
















1100 
















1 




/ 










/ 


























i 






j 


1 


1 








/ 


















1000 










/ 






I 












/■ 






























/ 






1 












/■ 


















90( 


) 










/ 






1. , 


1 








































f 




/ 


— ^ 






























jot 


) 










/ 


















































/ 






























1 


700 












A 


// 










7 














1 ^x 


ix 




f 












i 


J 


1 


7 










/ 












XI 


V 








60 


) 








K 


/ 


1 


/ 










/ 










> 


























// 








'/ 






t' 


n^ 


Y 














50( 














i 










/ 






X 


] 
























A 




1 


f 








/ 




A 


1 




















40C 










iJ 










\ 


^ 


] 




























/ 


1 


1 










■A 


r^ 


I 
























300 




ll 




/ 








^ 


r 
































n 




' 




X,„ 


<^\ 


y 






ALL BARS-1-3 ATLAS 

4''! 2 HlGH-4'00 WIDE 

SPAN FOR B 36" 

" B^ 16'' 

" B2 16'' 














20( 




(/ 


J 






k^ 


P ^- 






















/ 




/ 


^ 


C 






















m 


,( 


A 


V' 




--0- 


p^ 




















/\A 


V 


_^^ 


H 




'deflections of each bar 
left shows sets whe 


ARE REPRESENTED BY 2 CUR 
N LOAD AT GIVEN POINT IS Er 


VES; THE ONE TO THE 
JTIRELY REMOVED. 






<0 


03 




.0 


ce 


.008 


1 
.010 


1 
.012 


1 

.014 


1 
.016 


.ols 


.020 









.001 .002 .004 



DEFLECTION AT CENTERJN INCHES 



584 



BENDING OR FLEXURE. 



ICh. XII. 



natural cement should be used at all where concrete or 
mortar may be subjected to flexure. 



Table XII. 
BRICK-MASONRY BEAMS. 
(Age of beams about equally 5 months, 8 days, and 6 months.) 

ROSENDALE-CEMENT MORTAR: I C. 2 S. 



Span, Inches. 


/ 

d' 


Stress in Extreme Fibre, Pounds per 
Square Inch. 


No. of Tests. 




Max. 


Mean. 


Mm. 




96 

78 
66 
42 


7-4 
6 

5-1 
3-2 


67 

81 
91 


54 
18 
56 
73 


38 

23 
54 


5 

4 
8 


PORTlvAND-CEMENT MoRTAR: I C, 3 S. 


96 

66 

42 


7.4 
5-1 
3-2 


173 
145 
229 


144 
120 
166 


124 
96 
94 


4 

4 

10 



Table XII exhibits some interesting results of the tests 
of brick-masonry beams. These investigations were made 
by Messrs. A. W. Gill and Frederick Coykendall, gradua- 
ting students in Civil Engineering in Columbia University 
in 1897. Fig. I shows the manner of laying up the brick 
to form the beams which were tested. The breadth of 
each beam was about 12 ins. and the depth 13 ins. The 
spans varied from 8 ft. down to 3 ft. 6 ins., with the 
ratios of length over depth of beam given in the column 

headed -j. This column of ratios shows that the beams 
a 

should be considered short. 

The Rosendale-cement mortar was mixed with one vol- 



Art. oi.] 



BRICK-MASONRY BEAMS. 



585 



ume of cement to two volumes of sand, while the Portland- 
cement mortar was mixed with one volume of cement to 
three volumes of sand. During the first three months the 
beams were kept well wetted, but less so during the last 
three months. At no time were they dry. The Table gives 




Fig. I. 

all the results of tests and shows that the beams had very 
little resisting capacity, although possibly 15 to 20 pounds 
per square inch might be justified as working values in the 
extreme fibres of the beams built with Portland-cement 
mortar. The bricks were laid by ordinary masons with 
such care as could be impressed upon them, although the 
experimenters stated that the brickwork was of very in- 
different quality and hence that the results are lower than 
they should be. 



586 



BENDING OR FLEXURE. 



[Ch. XII. 



Natural-stone Beams. 

Table XIII exhibits results found by the same experi- 
menters as in the case of Table XII with a number of 
natural-stone beams, the spans for which varied from 36 
ins. down to 12 ins. The first figure in the second column' 
'of the table headed " Section " gives the depth of each beam, [ 

Table XIII. 
NATURAL-STONE BEAMS. 

BLUESTONE. 









Stress in Extreme Fibre, 




Span, 


Section, 


/ 


Pounds per Square Inch. 


No. of 


' Inches. 


Inches. 






Tests. 


d 


Max. 


Mean. 


Min. 


24 


4X6 


6.15 


3,958 


3,512 


3,054 


5 


36 


6X8 


6.2 


3,288 


2,797 


2,906 


3 


12 


4X6 


3 


4,112 


3,237 


2,282 


II 


24 


8X6 


3 


3,929 


3,547 


2,715 


6 



GRANITE. 



24 


4X6 


6 


2,321 


2,250 


2,178 


3 


36 


6X8 


6 


1,861 


1,798 


1,766 


3 


12 


4X6 


3 


2,714 


2,487 


2,086 


9 



SANDSTONE. 



24 


4X6 


6 


1,575 


1,354 


1,237 


3 


36 


6X4 


6 


1,204 


945 


637 


3 


12 


4X6 


3 


1,907 


1,539 


1,267 


9 



MARBLE. 



24 


4X6 


6 


2,036 


1,880 


1,617 


3 


36 


6X8 


6 


1,683 


1,548 


1,354 


3 


12 


4X6 


3 


2,455 


2,026 


1,696 


9 



Art. 91.] NATURAL-STONE BEAMS. 587 

while the second figure gives the width. It will be observed 

from the ratios of -1 given in the third column that the beams 

were very short. The extreme fibre stresses are seen to run 
comparatively high for the bluestone, granite, and marble. 
Indeed, working values of intensities may reasonably be 
taken as follows: 

For blue tone 250 to 400 pounds per square inch. 

' * granite 200 to 300 " " * * 

" marbl 17 to 225 " " " * 

" sandstone , 100 to 150 " " " ** 

In the use of sandstone it should be understood that the 
preceding \alues apply only to the best quaHties of that 
particular stone. 



CHAPTER Xril. 

CONCRETE-STEEL MEMBERS. 

Art. 92. — Composite Beams or Other Members of Concrete 

and Steel. 

Concrete, like other masonry, is admirably adapted to 
resist compression. Its capacity of resistance to tension 
is much less than its ultimate compressive resistance, 
although if the concrete is well made the tensile resistance 
may have considerable value. The purpose of the con- 
crete-steel combination is the production of a beam or 
other member almost entirely of concrete, but which shall 
have a high capacity to resist tension in those portions 
which may be subjected to tensile stresses. This result 
is accomplished by embedding steel bars of desired shape 
and of suitable cross -sectional area in the proper parts of 
the concrete. While no general rule can be given for the 
area of the steel section in comparison with the concrete, 
it may be stated approximately that the steel section is 
usually between f and r^ per cent, of the area of a normal 
section of the concrete. Inasmuch as the presence of the 
steel is for the purpose of giving tensile resistance to the 
member it is evident that the re-enforcing steel bars wih 
always be found in those portions of the concrete mass 
which may be subjected to tension. In such concrete- 
steel construction as arches the steel re-enforcement is 
frequently used both on the tension and compression sides 
of the concrete. 

588 



Art. 93-] CONCRETE-STEEL BEAMS. 589 

In the case of concrete-steel beams or other similar 
members, as the steel is entirely embedded in the concrete, 
the loads and reactions must obviously be applied directly 
to the latter. When the concrete takes its stress, there- 
fore, at "least a portion of that stress must be conveyed to 
the steel, and that requires that the adhesive joint or bond 
between the steel and concrete shall be as strong as possible. 
Hence in laying the steel bars in the concrete it is necessary 
that the contact between the two materials shall be inti- 
mate and essentially continuous. Various means are em- 
ployed to accomplish these ends. Square bars are fre- 
quently twisted, while round bars may be nicked and fiat 
ones either twisted continuously in one direction or have 
alternate portions twisted in opposite directions, or, finally, 
rolled with alternately enlarged and contracted sections. 
Again, where built-up members are embedded in concrete, 
rivet-heads and other details of construction serve the 
same general purposes. The efficiency of the concrete- 
steel construction depends wholly upon the resistance of 
this bond, and the design must always be such that the 
adhesive shear, so to speak, or the stress of sliding along 
the steel surface, shall never exceed per square unit the 
ultimate resistance of the bond. 

In the analysis and computations which follow it is 
assumed, as it must be, that the bond between the steel 
and concrete is such as to make the entire mass act as a 
unit, so that the combination of the two heterogeneous 
elements shall act as a single whole. 

Art. 93.— Physical Features of the Concrete-steel Combination 

in Beams. 

It will be shown later on that so far as can be deter- 
mined from physical data now available the coefficient of 
elasticity for concrete in compression for the operations 



^^o CONCRETE-STEEL MEMBERS. [Ch. XIII. 

ordinarily employed in designing engineering structures 
and for mixtures not less rich in cement than i cement, 
3 sand, and 6 gravel or broken stone, at ages of one to six 
months, may range from about 2,000,000 pounds per square 
inch to more than 4,000,000 pounds per square inch, while 
for concrete beams the coefficient or modulus may range 
from about 1,500,000 pounds per square inch for compara- 
tively shallow beams to more than 3,000,000 pounds per 
square inch for beams of comparatively great depths. 
Values for the coefficient of elasticity for concrete in 
tension can be found in Art. 60. Further tests for the 
determination of this quantity are much to be desired, but 
enough has been done to estabHsh at least closely approxi- 
mate values. Some authorities assume the tensile coeffi- 
cient to be much less than the coefficient of elasticity for 
concrete or mortar in compression. As a matter of fact, 
the tests of a Monier arch of 75 feet span by a committee 
of the Austrian Society of Engineers and Architects, 
which made its report in 1895, showed in that particular 
case the coefficient of elasticity of concrete in tension to be 
nearly one fifth greater than the coefficient for compression, 
although it should be stated that the age of the tensile 
specimens was materially greater than that of the com- 
pression material. The values in Art. 60 indicate that the 
tensile coefficient is at least equal to the compressive. 
It is possible that subsequent investigations may show 
that the tensile coefficient of elasticity is less than that for 
compression, but at the present time there appears to be 
practically no basis for that assumption. It seems to 
be reasonable and safe, as it is more simple to take 
the two coefficients equal to each other until further 
investigations have conclusively established a different 
ratio. 

It is important to state in this connection that the re- 



Art. 93-] CONCRETE-STEEL COMBINATION IN BEAMS. 591 

suits of tests with concrete-steel beams, so far as they have 
been made, indicate that the elastic or semi-elastic behavior 
of concrete under stress will in the main characterize the 
behavior of the same material when under loading in the 
composite beam of concrete and steel, so that the coefficients 
of elasticity determined for concrete alone may be used in 
the composite member. 

There is one important respect in which the action of 
concrete alone is quite different from that which takes place 
when it is combined with steel. In the latter case the con- 
crete will stretch under a stress nearly or quite equal to its 
ultimate resistance a comparatively large amount. It is 
sometimes stated that under such conditions the coeffi- 
cient of tensile elasticity of the concrete is practically zero, 
but there is just as much ground, or more, for making the 
same observation in connection with such ductile materials 
as structural steel. What is actually meant is simply that 
the concrete will stretch before parting much more when its 
deformation is controlled by the corresponding deformation 
of the steel reinforcement than when it acts by itself or 
without such reinforcement. This feature of the action 
under stress of concrete in the composite beam has a most 
important bearing upon som.e rather peculiar phenomena 
connected with the testing of such beams to failure. M. 
Considere has stated (' ' Comptes Rendus Academic des Sci- 
ences, ' ' Paris, Dec. 12, 1898) that mortar will stretch twenty 
times as much when combined with steel as when tmaided 
by that combination. He further states that the concrete 
stretches uniformly with uniform increments of bending 
moment up to about four tenths of the ultimate moment. 

As the coefficient of elasticity for concrete is a small 
fraction only of that of steel the tendency of the concrete 
in composite beams is to stretch or compress more than 
the steel embedded in it. Hence the concrete immediately 



592 



CONCRETE-STEEL MEMBERS. 



[Ch. XIII. 



adjacent to the steel tends to slide along the latter, but 
that tendency is resisted by the adhesive shear at the joint, 
in consequence of which the steel acquires its stress whether 
of tension or compression. The normal section of the 
unloaded beam, therefore, will not remain normal after 
flexure, but there will be either a cup-shaped depression 
around the steel or a similar shaped elevation. This is 
illustrated in Fig. i. 



i^^ 



^S 



Fig. I. 

In that figure the intensity of stress on either side of 
the neutral axis is assumed to var}^ directly as the distance 
from the axis, but in a subsequent analysis a different law of 
variation will be assumed in order that the treatment may 
be complete, although the author is not of opinion that the 
assumption of any law of variation different from that of 
the common theory of flexure is at the present time justified. 
It will further be assumed in the analysis which follows 
that normal sections of the unloaded beam will remain 
normal under loading. This is a common procedure, and 
it is not believed that the amount of variation from a plane 
section under stress, described above, is sufficient to make 
the assumption sensibly in error. 



Art. 94 —Rate at Which Steel Reinforcement Acquires Stress. 

The determination of the rate at which the concrete 
gives stress to the steel is not of great importance in ordi- 
nary design work or in most other practical relations; yet 



Art. 94-] R^TE OF ACQUIRING STRESS. 593 

it is desirable in some cases, and it is an element of the 
action of internal stresses in a composite beam which 
should be understood as clearly as practicable. The fol- 
lowing analysis offers a means of determining that rate 
as nearly as it can be done at the present time. The 
notation used is shown also in Fig. 3 on the opposite page. 
The intensity of stress in the concrete at the distance 
d^, the distance of the steel reinforcement, from the neutral 
axis is k. Then if / represent the moment of inertia of the 
entire composite section about its neutral axis (located by 
c/j, determined hereafter), there may be written 

^^ kl ,^ ^ dk .1 

^-T^' ■■■'^-^- ■ • • ■ W 

If 5 is the total transverse shear in the normal section 
in question at the distance x from one end of the beam, 

dM=Sdx = ^^ (2) 



d 



Let p be the total perimeter of section of the steel re- 
inforcement at the section located by x. 
Let A^ be the area of steel section with perimeter p. 
Let 5^ be the intensity of adhesive shear at the surface or 

. joint between the steel and concrete. 
Let k^ be the intensity of stress in the steel. 

The variation of k^ for the indefinitely small distance 
dx is dk^. From what has preceded there may be written 

i>.dx.s'=A,dk,; ,.dx^^. .. . . (3) 

Inserting the value of dx from eq. (3) in eq. (2), 

ps' d. 



594 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

By solving this equation for s' and remembering that 

dk2 _E2 

dk El 

, c^ A do dko c-^2G?2v42 / X 

Ip dk El I p 

This value of s' must never exceed the ultimate adhe- 
sive resistance between the steel and concrete. 

Tests for the determination of the adhesive shear between 
concrete and imbedded round rods have been made by Pro- 
fessors Talbot, Withey, Hatt, Duff A. Abrams and others. 
In view of the inevitable uncertainties of condition of such 
rods in respect to the bond between them and the concrete, 
greatly varying values must be anticipated, as they will 
depend upon the age proportions of the concrete, the smooth- 
ness (or roughness) of the surface of the rods, the amount 
of water used in mixing the concrete and the continuity of 
contact between the concrete and the rods. The value of 
adhesive shear has sometimes been taken as i6 to 20 per 
cent, of the ultimate compressive resistance of the concrete, 
but this is probably too high, even for the best qualities of 
concrete. 

Again, the ultimate value of adhesive shear as deter- 
mined by the pulling of rods directly from a block of con- 
crete may be materially different from that developed in 
a bent beam and, hence, the latter procedure should be the 
basis of determinations for reinforcing rods for beams. A 
clear distinction should be drawn between the adhesive 
shear existing prior to movement of the rod in its mastic 
and the resistance to that motion after it once begins. 

Professor M. O. Withey published in a Bulletin of the 
University of Wisconsin, No. 321, 1909, the data of a large 
number of tests in which the results were obtained from 



Art. 94.] ADHESIVE SHEAR OR BOND. 595 

loaded beams, the stretch of the rods being accurately 
measured by an extensometer for a given length of imbedded 
rod. The diameter of rod was f inch and the age of the 
concrete varied from seven days up to six months. A large 
number of tests gave the adhesive shear as varying from 
a minimum of 129 pounds per square inch to a maximum 
of 362 pounds, a few only of the results falling below 200 
pounds per square inch. It would probably be fair to 
take 250 pounds per square inch as a representative 
average of these results. 

In a series of tests with diameters of bars running from 
f inch to I inch, the average results were 278 and 286 
pounds per square inch for the two smaller sizes of bars 
and 163 pounds and 195 pounds per square inch for the 
I -inch bars. The age of the 1-2-4 concrete in this case 
was two months. 

There may be found in Bulletin No. 71, University of 
Illinois, a full account of a large number of " Tests of Bond 
between Concrete and Steel," by Duff A. Abrams. These 
tests were made under a great variety of conditions as to 
age, sizes of rods, surface of rods, i.e., whether plain or 
deformed, shapes of cross-sections, rods pulled out of blocks 
and rods stressed in reinforced concrete beams, accompanied 
by extended observations as to effects of loading including 
careful measurements of the stretch of steel both in pulling 
rods from blocks and as they were stressed in beams. In 
these tests a clear distinction was recognized between the 
adhesion to the surface of the rods and the resistance of 
movement after initial slip, the greatest intensity of bond 
resistance visually being developed after the beginning of 
slip. 

A roughened surface of rod will obviously yield a greater 
bond resistance than a perfectly smooth surface, the resist- 
ance of the latter being almost wholly adhesion. 



50 



CONCRETE-STEEL MEMBERS. 



[Ch. XIII. 



The following are a few of Mr. Abrams' conclusions : 
" (41) The mean computed values for bond stresses in 
the 6 -foot beams in the series of 191 1 and 191 2 were as 
given below. All beams were of 1-2-4 concrete, tested at 
2 to 8 months by loads applied at the one third points of 
the span. Stresses are given in pounds per square inch. 



I and I j-in. plain round. 

|-in. plain round 

|-in. plain round 

I -in. plain square 

I -in. twisted square. . . . 
i^-in. corrugated round . 



Number 


First End 


End Slip 
of o.ooi In. 


of Tests. 


Slip of Bar. 


28 


245 


340 


3 


186 


242 


3 


172 


235 


6 


190 


248 


3 


222 


289 


9 


251 


360 



Maximum 
Bond Stress 



375 
274 

255 
278 

337 
488 



" (42) In the beams reinforced with plain bars end slip 
begins at 67 per cent, of the maximum bond resistance; 
for the corrugated rounds this ratio is 51 per cent., and for 
the twisted squares, 66 per cent. 

" (43) The bond unit resistance in beams reinforced 
with plain square bars, computed on the superficial area of 
the bar, was about 75 per cent, of that for similar beams 
reinforced with plain round bars of similar size. 

" (44) Beams reinforced with twisted square bars gave 
values at small slips about 85 per cent, of those found for 
plain rounds. At the maximum load, the bond-unit stress 
with the twisted bars was 90 per cent, of that with plain 
round bars of similar size. 

" (45) In the beams reinforced with i|^-inch corrugated 
rounds, slip of the end of the bar was observed at about 
the same bond stress as in the plain bars of comparable 
size. At an end slip of 0.00 1 inch, the corrugated bars gave 
a bond resistance about 6 per cent, higher and at the maxi- 
mum load, about 30 per cent, higher than the plain rounds. 



Art. 94.] ADHESIVE SHEAR OR BOND. 597 

" (46) The beams in which the longitudinal reinforce- 
ment consisted of three or four bars smaller than those used 
in most of the tests gave bond stresses which, according to 
the usual method of computation, were about 70 per cent, 
of the stresses obtained in the beams reinforced with a sin- 
gle bar of large size." 

As the greatest bond stress was developed after the 
beginning of slip, the preceding results show that such a 
maximum value exists beyond a net slip of 0.00 1 inch. 

Again referring to the resistance of deformed bars, he 
states, " The mean bond resistance for the deformed bars, 
tested was not materially different from that for plain 
bars until a slip of about .01 inch was developed; with a 
continuation of slip, the projections came into action and 
with much larger slip high bond stresses were developed." 

Again referring to a working bond stress, he states : 

" (59) A working bond stress equal to 4 per cent, of ths 
compressive strength of the concrete tested in the form of 
8- by 16-inch cylinders at the age of 28 days (equivalent to 
80 pounds per square inch in concrete having a compressive 
strength of 2000 pounds per square inch) is as high a stress 
as should be used. This stress is equivalent to about one 
third that causing first slip of bar and one fifth of the maxi- 
mum bond resistance of plain round bars as determined 
from pull-out tests. The use of deformed bars of proper 
design may be expected to guard against local deficiencies 
in bond resistance due to poor workmanship and their 
presence may properly be considered as an additional safe- 
guard against ultimate failure by bond. However, it does 
not seem wise to place the working bond stress for deformed 
bars higher than that used for plain bars." 

The preceding results were obtained from statically 
loaded beams. Professor Withey found no injurious effects 
on the resistance of adhesive shear under repeated loads until 



598 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

the latter became 50 to 60 per cent, of the ultimate static 
loads. This last percentage may be raised to 60 to 70 per 
cent, with corrugated bars. Investigations made by the 
same authority indicate that the results of static tests on 
smooth round rods imbedded in beams will give values 
for the bond or adhesive shear between the concrete and 
the rods from one half to two thirds only of corresponding 
results obtained by pulling imbedded steel rods from the con- 
crete cylinders, but Mr. Abrams appears to believe that 
the results of properly made ' ' pull-out ' ' tests will be about 
the same as found for beams. 

While materially larger values for ultimate resistance 
of adhesive shear have been reported by some experimenters 
with small rods, it appears prudent not to take the ultimate 
resistance greater than perhaps 200 to 350 pounds per square 
inch for round or square rods from ij inch to f inch in 
diameter. 

The working value for this bond for adhesive shear 
should not be taken more than one fourth to one fifth of 
its ultimate value. 



Art. 95. — Ultimate and Working Values of Empirical Quan- 
tities for Concrete-steel Beams. 

It is necessary for the practical use of the preceding 
and following analyses that a number of empirical quanti- 
ties be determined, chiefly for the concrete. The coeffi- 
cient of elasticity for wrought iron for this purpose may 
be taken at 28,000,000 pounds per square inch, and 
30,000,000 pounds per square inch for structural steel, 
which is now generally used in the reinforcement of con- 
crete-steel beams. 

The modulus of elasticity for concrete at different ages 
and for different proportions of matrix and aggregate has 



Art. 95-] ULTIMATE AND WORKING VALUES. 599 

been fully considered in Art. 67, and Table I of that Article 
exhibits a full set of values. A mixture of i cement, 2 sand 
and 4 broken stone or gravel is generally used in rein- 
forced concrete work ; and for such concrete the Table cited 
above shows that the modulus of elasticity at the age of 
one month may be taken from about 1,500,000 to nearly 
3,000,000. In view, however, of the uncertain conditions 
attending the making of concrete on actual work a higher 
value than 2,000,000 is seldom used. The ratio of the 
modulus for steel divided by that for concrete is generally 
taken at 15, although 12 is sometimes employed, the latter 
value implying a modulus for concrete of 2,500,000. 

The ultimate resistances of mortar and concrete in 
tension and compression will be found in Arts. 60 and 67.- 
These values will also depend upon the proportions and 
character of mixture or upon the age. The records of 
tests and experience which have thus far accumulated in 
connection with concrete-steel construction show that the 
compressive working stress of concrete in beams, where the 
mixture is in the proportions of i cement, 2 sand, and 4 
gravel or broken stone, may probably be taken as high as 
500 pounds per square inch. It should be remembered 
that this intensity will exist in the extreme fibres of the 
beam only. Mixtures of less strength would require a 
corresponding reduction in the maximum working in- 
tensity of compression. A mixture, for example, of i 
cement, 2 J sand, and 5 broken stone, unless the materials 
were well balanced, might justify a reduction of the" 
greatest working stress to 400 pounds per square inch. 

Some foreign authorities have prescribed two degrees 
of safety, in the first of which the maximum working stress 
of compression of 427 pounds per square inch is allowed, 
and 711 pounds per square inch for safety of the second 
degree. Structures in which the duty of the concrete is 



6oo CONCRETE-STEEL MEMBERS. [Ch. XIII. 

severe might be designed with the smallest of those values, 
but where the duty is materially less severe, with the 
larger. 

It is not unusual at the present time in the design of 
concrete-steel arches to allow a maximum mtensity of 
compression of 500 pounds per square inch and 50 to 75 
poimds per square inch for the maximum intensity of 
tension, if tension is allowed. 

Tensile tests of concrete show that where proportions 
of I cement, 2 sand, and 4 gravel or broken stone are used 
a maximum intensity of tension of 50 to 70 pounds per 
square inch is about J to ^ the ultimate tensile resistance 
at the age of three to six months. These values are reason- 
able and may be employed in concrete work where it is 
permitted to avail of the tensile resistance of concrete. In 
much of the best engineering practice at the present time, 
however, the tensile resistance of the concrete is neglected 
in the interests of additional safety in concrete-steel beam 
construction. Inasmuch as fine cracks may appear in 
concrete from other agencies than tensile stress, it is un- 
doubtedly advisable in most cases certainly to omit the 
bending resistance of the concrete in tension, especially 
as that omission does not sensibly increase the weight or 
cost of the beam when properly designed. 

Art. 96. — General Formulae and Notation for the Theory of 
Concrete-steel Beams According to the Common Theory of 
Flexure. 

The application of the common theory of flexure to 
the bending of concrete-steel beams is in reality the de- 
velopment of the theor}^ of flexure for composite beams 
of any two materials. The notation to be used and the 
general formula will first be written, therefore, and then 
the special formulae for concrete-steel beams will be estab- 



I 



Art. 96.] GENERAL FORMVLJE AND NOTATION. 6oi 

lished in the succeeding articles. These general formulae, 
it should be observed, apply to beams of any shapes of 
cross-section of either material or for any relative areas 
of cross-section of those materials, although in concrete- 
steel beams the area of cross-section of the steel is frequently 
or perhaps usually but one to one and a half per cent, of 
the area of the concrete. 

Again, the formulae will be so written as to make 
practicable the use of different coefficients of elasticity 
for concrete in tension and compression if that should 
be desired. 

The notation to be used in the succeeding articles is 
chiefly the following: 
E^ = coefficient of elasticity of the steel. 
E^= " " " '' '' concrete in compression. 

nE^= ** " " '' '' concrete in tension. 

A^ and A^ are the areas of normal section of the concrete 

and steel respectively. 
7j and I^ are the moments of inertia of A^ and A^ respec- 
tively about the neutral axis of the normal section. 
k^ = greatest intensity of bending compression in the con- 
crete. 
^' = greatest intensity of bending tension in the concrete. 
c = greatest intensity of bending compression in the steel. 
/ = greatest intensity of bending tension in the steel. 
6= breadth of the concrete. 

h and /i/ are total depths of the concrete and steel re- 
spectively. 
7^2= vertical distance between the centres of the steel 

reinforcing members. 
Jj= distance of extreme compression "fibre" of the con- 
crete from the neutral axis. 
(^2 = distance of the centre of the compression steel rein- 
forcing member from the neutral axis. 



6o2 CONCRETE-STEEL BEAMS. [Ch. Xlll. 

ds = distance from the neutral axis to the centre of 

the tension steel reinforcement. 
d^ = distance from extreme compression fibre of the 
steel to the neutral axis, 
a = distance of the centre of the compression steel 
reinforcing member from exterior compression 
surface of concrete. 
ai = distance of the centre of the tension steel rein- 
forcing member from exterior tension sur- 
face of concrete. 
rA2=area of normal section of reinforcing steel in 
tension. 
(1—^)^2= area of normal section of reinforcing steel in 
compression. 
Jb= intensity of compressive stress in the concrete 
at distance z from the neutral axis. 
^''= intensity of tensile stress in the concrete at dis- 
tance ^ from the neutral axis. 
^2 = intensity of stress in the steel at distance z from 

the neutral axis. 
w= tensile or compressive strain in unit length of 
' ' fibre ' ' at unit distance from the neutral 
axis. 
In all the theory* of bending of concrete-steel beams it 
is assumed, as in the common theory of flexure, that any 
plane, normal section of the beam, before bending takes 
place, will remain plane (and normal) while the beam is 
subjected to bending. Hence 

k=Eiuz, k" =nEitiz, and k2=E2UZ. . (i) 

Inasmuch as all the loading carried by concrete-steel 
beams is supposed to act in a direction normal to the axes 

* Given in Art. 32. Eqs. (i) to (4) are simple adaptations of the equa- 
tions of that Art. to this case. 



Art. 96.] 



GENERAL FORMULA AND NOTATION. 



603 



of the beams, as is usual in the common theory of flexure, 
the total stresses of tension and compression in any normal 
section of a beam induced by the bending must be equal 
to zero. The expression of this sum, written by the aid of 
eqs. (i) and by which the neutral axis of the composite 
section is determined, is the following: 



E.n[£^zdA,^nfl^zdA,] ,-E,u£_^jdA,=o. 



(2) 



Or 



/ zdA^^n zdA.+j^ ,,^dA,= 



o. . (3) 




Eq. (3) is perfectly general, and the position of the 
neutral axis can always be located by it whatever may 

be the shape of cross-section 
of either the concrete or steel. 
Fig. I may be taken as 
an arbitrary typical com- 
posite section showing the 
preceding system of notation 
applied to it. The outline 
Fig. I. of the concrete is rectangular, 

as in the ordinary concrete-steel beam. The steel in com- 
pression is represented as tAvo steel angles, while three 
round rods constitute the steel in tension. In the next 
article the application of the general eq. (3) to the special 
case of the ordinary concrete-steel beam will be made. 

The general value of the bending moment of the stresses 
induced in any normal section of a composite beam can be 
at once written by the aid of eqs. (i). The typical ex- 
pression of the differential moment is 



kdAiZ=E^uzHA^, 



6o4 CONCRETE-STEEL BEAMS. 

Hence the value of the moment is 



[Ch. XIII. 



M = E,uf\^dA, + nE,nfl^z^dA, + E,u£_^^z^dA,. (4) 

This equation is also completely general whatever may 
be the shape of section of eitner material. It will be de- 
veloped for the ordinary form of concrete-steel beams in 
Art. 97. 

Eqs. (3) and (4) cover completely the theory of bending 
or flexure of composite beams of two materials, one of 
them having different values for the coefficients of elasticity 
in tension and compression. It will be observed that the 
position of the neutral axis of any section of the beam, as 
located by eq. (3), is affected by the values of E^, E^, and n^ 
and that it does not in general pass through the centre of 
gravity of the section. 

Art. 97, — T-Beams of Reinforced Concrete. 

The general formulae of Art. 96 belong to beams of any 
shape of cross-section whatever; it is only necessary, there- 



'( 



-^-O-G-O- 



FlG. I. 



fore, in this case, to apply them to the T-shaped section. 
Two conditions, may arise, in one of which the neutral 



Art. 97.] T-BEAMS OF REINFORCED CONCRETE. 605 

axis lies in the flange of the beam whose cross-section is 
shown in Fig. i, or, as shown in that figure, it may He below 
the flange. As is usually the case in actual work, the 
tensile resistance of the concrete will finally be neglected. 
This latter condition makes it necessary to consider only 
the case shown by Fig. i. 



Position of Neutral Axis. 

Using the notation of Art. 96 under the conditions out- 
lined above, but first recognizing the tensile resistance of 
the concrete, 

rdi rdi rdi-f 

I zdAi= I Z'b^dz-i- j Z'bdz 

Jo Jdi-f Jo 

=w(«.-f)+i^^' w 

Again, ^ 

X^ r^ nb 
zdAi=ni zbdz = (hi^ — 2hidi+di^). (la) 
- di Jhi- di 2 



As the steel section is small it will be essentially correct 
to consider each part of it concentrated at its centre of 
gravity. Hence there may be written, 



rd,' 

1 zdA2 = (i -r)A2d2-rA2{n2 -d2) =^2(6^2 -r/zs). {ib) 

JW-d^ 



Introducing the values given by eqs. (i), (la) and (2) 
in eq. (3) of Art 96, 



6o6 CONCRETE-STEEL BEAMS. [Ch. XIII. 

bHd. -b^'- + — ^ bdA + --- -— ' h nbh.d, - n — ^ 

2 2 2 2 2 

+ 1^.4, (d, -f/;,)=o. 



I — n 1 — n 

+ ,E,A, (a+rh,) 

E^ b 1 — n 

The solution of this quadratic equation will give 



1—71 



3 

If the two coefficients of elasticity for concrete in tension 
and compression are the same, as is always assumed in 
actual work, n = i. This value gives indetermination in 
Eq. 3, but it is only necessary to multiply both members 
of Eq. 2 by (i — n) and then make n = i. These opera- 
tions give 

d, = 



'if -h '••*¥. 



E,A 



If the entire steel reinforcement is on the tension side of 
the beam, r = i, and in Eqs. 3 and 4, a + rh^ =a-\-h2=hi 
The tensile capacity of the concrete is practically always 



Art. 97.] T-BEAMS OF REINFORCED CONCRETE. 607 

neglected; hence n = o in Eq. 3, and 



.,=-/!-.) 



E, b 



,*v/(r--)'--l;t(°-'-^('C---)-ftt)' s 

These formulce locate the neutral axis by giving the dis- 
tance d^ for all cases. 

An important special case arises where the neutral axis 
NS, Fig. I, lies in the lower side of the flange, i.e., when 
di =/. Making that substitution in the equation preceding 
eq. (2), 

j,2^_JI^+^^(,5/.,+f ^,) =^l^+^A,(a + rh,). (6) 

2 \ iLi / 2 rLi 

Solving this quadratic equation, 

nbhi+-=^A2 
^1 = 5'-n6 . 



Inbhi+^A2\ nbhi^ + 2^A2{a+rh2) 

^V\-T^3S-/+ fe^ • • (7) 

If concrete in tension be neglected, n =0 and, 



, E2A2^ (E2A2Y, E2A2{a+rh2) .^. 

Eq. (8) shows that the case of a T-beam with neutral 
axis at the lower surface of the flange and with tensile 
resistance of concrete neglected is equivalent to a solid 
rectangular beam of the same width as the flange under 



6o8 CONCRETE-STEEL BEAMS. [Ch. XIII. 

the same assumption of the neglect of the concrete in ten- 
sion. No material error will be committed in assuming 
any T-beam similarly equivalent to a solid rectangular 
beam if the neutral axis is near the under side of the flange. 
If the neutral axis NS lies in the flange the area (b' — b) 
(f—di) of concrete flange section will be in tension. In 

that case the term —n{b'—b)— must be added to the 

2 

third member of eq. (la), and hence to the first member of 
the equation preceding eq. (2). This will add obvious cor- 
responding terms to eqs. (3), (4) and (5), but the special 
case is so rare that it needs no further attention. Unless 
{f—di) has material value eqs. (7) and (8) may be used. 

Balanced or Economic Steel Reinforcement. 

In order that there may be economy of material it is 
necessary that the relation between the cross-sectional areas 
of the steel and concrete may be such as to make the 
greatest intensities of stress in each equal to the prescribed 
working stresses. This condition is said to make a " bal- 
anced " section or a balanced percentage of steel reinforce- 
ment. 

' In the general case of tensile and compressive steel 
reinforcement with the tensile resistance of concrete recog- 
nized, the equality of the total tensile and compressive 
stresses in a normal section of a T-beam gives eq. (9), if 
the neutral axis lies in the under surface of the flange, as 
is assumed in establishing eqs. (6), (7) and (8); 

^kidib'^c{i-r)A2^l^d^bdz+rA2t. . . (9) 

Adding \k\b'dz to each side of eq. (9) and then dividing the 
resulting equation by b'{d\-\-dz) =b'hi, eq. (10) will result: 



Art. 97.] T-BEAMS OF REINFORCED CONCRETE, 609 



or 



2 ^ c(i —r) —rt J 



(loa) 



It will now be convenient to simplify the forms of the 
preceding equations by using the following notation : 

e=-:-^, the ratio of the modulus of elasticity for steel 

over that for concrete. Usually e = i^, but occa- 
sionally ^ = 12. 

p =Yjr-, the steel ratio, usually expressed as per cent, of 

hi 

total rectangular section, i.e., in case of the T-beam 
per cent, of total rectangular outline h'hi. 
^1 n 



The steel ratio or per cent, p, is written in terms of the 
circumscribing rectangle b^hi in the interests of simplicity 
and as being at least as rational as any other method. 

The effective depth of the beam is taken as hi because 
the exterior thickness of concrete (h—hi) is usually a pro- 
tecting shell against fire, possibly to be partially or wholly 
destroyed in a conflagration, and, hence, not to be counted 
as effective beam material. The formulae may easily be 
changed so as to be expressed in terms of the full depth 
h by simply writing h—o for hi, o being the difference 
(h—hi), i.e., the thickness of the concrete from the centre 
of the tension steel reinforcement to the lower surface of 
the web or stem of the beam, usually 2 to 3 inches, or 



6io CONCRETE-STEEL BEAMS. [Ch. XIII. 

more for very large beams. The preceding notation will 
enable the following formulas for practical use to be written. 

FormulcB to Locate Neutral Axis in T-Beams. 

Dividing eq. (3) hy hi and writing -=;^ — -r^ for 7:^ ^ ; 

lLi £Li 





= q = 


f(b' 
hi\b 


\ 

-I +n 
/ 

I —n 


+-ep 










1 <A 


F 


-^)4+" b' 


a+rh2 


, \hAb 


\ b' y 

-ij+n+j-epj 


(11) 


^\ 


(i-n)hi 




{i-nY 


' 



Doing precisely the same with eq. (4) there will result 
for the usual condition of the two moduli for tension and 
compression being the same, but with tensile resistance of 
the concrete recognized : 

d, '(h?\b-v+'+r-hr'^) 

-=q=- FTTT \ D • • (12) 



^1 lK_r) ~.^b' 



h.\b v+^+r^ 



For the special case of neglect of the tensile resistance 
of concrete, eq. (5) gives, after dividing both sides by hi : 

hr'^=-hk-')-h'^ 



it 
1 



If the neutral axis lies in the under side of the flange, 



Art. 97] T-BEAMS OF REINFORCED CONCRETE. 6ii 

both sides of eq. (7) are to be divided by hi, and that equa- 
tion may then take the form : 



(14) 



Or, if concrete in tension be neglected, n=o and eq. (8) 
then gives 





7 / 

n+—ep 


b-" 


//-K 


. a^-rh2h' ^ 


[h") 


' b' 

b-"" 



f^=q=-ep^^e^P^ + ,^ep. . . (15) 

If the reinforcing steel is wholly on the tension side of 
the beam section r = i and a+rh2=a-\-h2=hi. Hence in 

eqs. (12), (13), (14) and (15), — — - = i, but no other change 

hi 

is needed. 

The value of the steel ratio or per cent, for the perfectly 

general case is given by eq. (10), by placing p =777^ in that 

b hi 

equation and then solving for p, which will give : 



ki \di h' /hi f .. 

^=7 c{.-r)-rt (^^) 



i \dib'^ J hi 
2 c{i—r)—r 

If concrete in tension be neglected, eq. (9) shows that 

ki 

—dz=ki=o in eq. (16), and that equation will then take 

di 

the form 

j^_ki hi = _^ ^1 / \ 

^ 2c{i-r)-rt~ 2 {c{i-r)-rt)hi ' ^'^^ 



6i2 CONCRETE-STEEL MEMBERS. [Ch. XIII 

If the reinforcing steel is wholly on the tension side of 
the beam section, r = i and c{i —r) —rt= —rt. Eq. (17) 
will then take the form : 

^ ki hi ki di , ^. 

^=i-^=7r,h; ('«^ 

The laws of the common theory of flexure give the fol- 
lowing relations : 

ki t I di eki ' . s 

-T=--r', or -r= — -y .... (19) 
di e ds' ds t ' ^ ^^ 

hence : 

di+ds eki+t hi f . 



Also: 



c d2 -i . ds , . 

- = --, and t=-TC (21) 

t US a2 



ds 



Placing the value of — from eq. (20) in eq. (18) : 
hi 



p=Ittt — \ (^^) 

— I— + I 

The area of the steel section A 2 can at once be found from 
p in all cases by simply writing : 

;p = 777^, and hence, A2=pb'hi. . . (23) 

ki 

In all these equations for locating the neutral axis of a 

section NS, Fig. i , the ratios — , /- and other similar quan- 

hi 



Art. 97.] T-BEAMS OF REINFORCED CONCRETE. 613 

titles depending on the dimensions of the cross-section will 
be known, at least tentatively. Indeed in making prac- 
tical applications of these equations it will in general be 
necessary to assign trial dimensions of the cross-sections of 
the beam if the application is made for the design of the 
latter. Such trial dimensions must be assigned by the aid 
of prior experience or other beams already designed for 
more or less similar conditions. After trial dimensions have 
been tested by actual computations for the assigned loads, 
such modifications or revision of these dimensions as may 
be necessary must then be made. 

If the neutral axis lies within the section of the flange, 
the changes in the preceding formulae already indicated for 
that case may be easily introduced, but the case is so rare 
that complete expressions for its treatment need not be 
written. If the tensile resistance of the concrete is neg- 
lected, the formulae for the special case, only, of the neutral 
axis lying in the under surface of the flange are needed, 
simply considering the depth of flange / as the distance from 
the upper flange surface to the neutral axis. In fact that 
special case will cover the great majority of T-beams with 
sufficient accuracy for practical purposes. 

The general value of the steel ratio or per cent, for a 
balanced section may be considered as given by eq. (16) 
even though the neutral axis does not lie in the under sur- 
face of the flange, at least as a reasonably close approxima- 
tion even when the position of the neutral axis is materially 
different from that supposition. In determining that ratio 
or per cent, ki, c and t must be considered as prescribed 
working values of those respective intensities of stress, the 
ratio between c and / being fixed by the distances of 
the steel reinforcements from the neutral surface. When 
the steel reinforcement is wholly on the tension side, as 
in the usual cases, ki and t are prescribed working stresses 



6i4 CONCRETE-STEEL MEMBERS. [Ch. IXII. 

for the concrete in compression and the steel in tension, 
respectively. 

Art. 98. — Bending Moments in Concrete-steel T-Beams by 
Common Theory of Flexure. 

The complete expressions for the bending moments of 
concrete-steel T-beams may now be written and their 
values for any particular case estimated, by introducing 
the notation already employed into ec^ (4) of Art. 95. 
The moment of inertia or integral in the last term of the 
second member of that equation takes the form : 



r 

Jh't -d' 



z^dA2 = {i-r)A2d2^-\-rA2d3^. . . (i) 



Referring to Fig. i of Art. 96, the other two moments 
of inertia in the first and second terms of the second member 
of eq. (4) of Art. 95 become: 

Jo 3 3 

f ^dA.=b^h^^ (3) 

J hi -d, 3 



Also, 



Eiu=^\ and E2U=y^Eiu=^~. . . (4) 
di El Eidi 



Introducing these values in eq. (4) of Art. 95, remem- 
bering that hi —di =/t2 —d2 =ds the bending moment M for 
a T-beam become : 

dil 3 .S 3 El J 



Art. 98.] BENDING MOMENTS IN CONCRETE-STEEL T-BEAMS. 615 

This equation is written in terms of one intensity of stress 
ki for convenience in computation, but it will be advisable 
sometimes to use other intensities, such as the greatest 
stress t in the tensile steel reinforcement. This can readily 
be done by the aid of the following relations based upon 
common theory of flexure, in addition to the relations 

shown in eq. (4) and remembering that -^=e. 



c eki t c .. k ki ,,^ 

eEiu=-z-=—= — . Also— =-7^. . . . (6) 
di ds ct2 CI3 cti 



These simple values will enable any intensity to be expressed 
in terms of any other. The greatest compression in the 
concrete, ki, and greatest tension in the steel, t, are those 
mostly required. 

If, as is usual, the two moduli oj elasticity of concrete in 
tension and compression are equal to each other, n = i in eq. 
(5), but no other change is needed. 

Neglect of Concrete in Tension. 

If the resistance to concrete in tension be neglected, 
n=o in eq. (5), and: 

In ordinary T-beams all steel reinforcement is in tension ; 
hence r = i and eq. (7) becomes: 



6i6 CONCRETE-STEEL MEMBERS. [Ch. XIII 

Special Case of Neutral Axis in under Surface of Flange. 
In this case {di —f) =o and eq. (7) will take the form: 
k 



M = ^^[^d,^+3e^{d2'{i-r)+ds'r)]. . . (9) 



If the steel reinforcement is wholly on the tension side r = i, 
as in eq. (8). 

This special case may, without material error, be con- 
sidered to include all T-beams for which (<ii — /) or (/— <ii) 
is relatively small. 

Art. 99. — Concrete Steel Beams of Rectangular Section. 

All formulas for reinforced concrete beams with rect- 
angular section may be written at once from those for 
T-beams by simply making b^ =b in the latter. A typical 
rectangular cross-section for the general case is shown in 
Fig. I, Art. 96, although in the usual case the steel rein- 
forcement is wholly on the tension side. 

FormulcB to Locate Neutral Axis in Beams of Rectangular 

Section. 

The general case requires eq. (11) of Art. 97. Making 
h' = b that equation becomes : 

di n+ep , r^z \ a-\-rh2 , (n-\-epy . . 
— =q= ±A^ \-2ep- ri- + . \, - (i) 

If the moduli for tension and compression are the same, 
as is invariably assumed in engineering practice, b = b' in 
eq. (12), Art. 97: 

- +ep — 

di 2 k] / X 



Art. 99-1 CONCRETE-STEEL BEAMS OF RECTANGULAR SECTION. 617 

If the tensile resistance of the concrete be neglected, 
the same substitution of b =b' is made in eq. (13) of Art. 97 : 



q^-ep±y]2ep^±^+e^p^. . . (3) 



When the reinforcing steel is wholly on the tension side 
r = i and a-{-rh2=a-{-h2=hi, hence: 

j^=q= -ep±V2ep-^e^p^ (4) 

hi 

This is the ordinary case. 

It will be observed that eq, (3) is identical with eq. (15) 
of Art. 97. 

A2 
The steel ratio or per cent., p =7^—, for the general case 

bki 

of balanced sections is given by eq. (16) of Art. 97 by 
making b = b': 



1 \di /hi 



^ ki \di ' Jhi f . 

^=T .(i-r)-rt ^5) 

When the tensile resistance of the concrete is neglected, 
the value of p given by eq. (17) of Art. 97 remains un- 
changed : 

hi hi , s. 

^=T.(i-r)-rt ^^) 

Eq. (18) of the same article gives p as it stands if the 
tensile resistance of the concrete is neglected, i.e., of r = i : 

^=-2-r (7) 



6i8 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

Eqs. (19), (20) and (21) of Art. 97 hold true for rect- 
angular sections and, hence, eq. (7) may take the form of 
eq. (22) of that article: 



f=i^^^~^ w 



^iV^^i / 



These values of the steel ratio p will form the basis of 
economical beam design. The working stresses ki for con- 
crete in compression and t for steel in tension will be pre- 
scribed in the specifications for the work to be done. 

Bending Moments for Rectangular Sections. 

The general value of the bending moment, M, i.e., for 
unequal moduli in tension and compression and with tensile 
resistance of concrete recognized, is given by eq. (5) of Art. 
98 after making h' =h.' 

M=|-4^+n^+M2(ci22(i-r)+^32r)l. . (9) 
^iL 3 3 J 

If it be desired to express this equation in terms of other 
intensities than ki, the following relations given in eq. (6) 
of Art. 98 will enable that to be done: 



cii as <^2 <^3 cii 



The moduli jor concrete in tension and compression are 
invariably considered equal, and in that case w = i in eq. (9), 
but no other change is required. 



Art. 99.] CONCRETE-STEEL BEAMS OF RECTANGULAR SECTION. 619 

Neglect of Concrete in Tension. 

The neglect of the concrete in tension is affected by 
making n=o in eq. (9) giving: 



M 



= ^[^^+eA2{d2Ki-r)-{-dsh)\. . . (11) 



The steel reinforcement is usually wholly on the tension 
side, i.e., r = i. Making this substitution in eq. (11) : 



k 
a 



^^(^ll,+eA,d,^ (12) 



All the preceding values of the bending moment M may, 
if desired, be expressed in terms of the steel ratio p by sub- 
stituting phhi for A 2. 

In the preceding equations the distance di of the neutral 
axis from the exterior compression surface of the beam is 
to be found from the appropriate formula for q of this article, 
since di =qhi. 

The preceding equations complete all that are necessary 
in the treatment of practical questions of design or of 
ultimate carrying capacity. 

In all the preceding analyses of Arts. (97), (98) and (99), 
the total depth h of either the T-beam or the solid rect- 

angular section may be used if desired by making p = -r-jr 

h 

A . 

or =77' but in that case in the equations for di the fraction 

on 



a-\-h2 



(when r = i) will occur, that fraction having values 



h 
varying from about .67 for floor slabs to .95, for beams of 

much depth instead of — - — = 1. It is rare, however, that 

such a form of equation will need to be used. 



620 



CONCRETE-STEEL MEMBERS. 



[Ch. XIII. 



Art. 100. — Shearing Stresses and Web Reinforcements in 
Reinforced Concrete Beams. 

In the case of reinforced concrete T-beams it will be 
assumed that the stem or web extending from the upper 
surface of the flange down to the centre of the tension steel, 
i.e., having the depth hi and the width h, will carry the 
whole transverse shear. In the solid rectangular section, 
the total sectional area less that part of it below or outside 
of the centre of the tension steel reinforcement will be 
assumed to resist transverse shear, i.e., the resisting area 
will be hhi as in the case of the T-beam. 

Fig. I represents a simple T-beam supported at each 
end Q and R, having steel reinforcement both in the flange 



* 



b a 



^^ 




C D 



R 



Fig. I. 



and in the lower or tension part of the beam. In order 
to illustrate fully the action of the shearing stresses in such 
a beam, the tensile resistance of the concrete may be recog- 
nized. If there were no steel reinforcement, the analysis 
of Art. 15 shows that in the case of a rectangular section 
the greatest intensity of either longitudinal or transverse 
shear exists at the neutral axis of the section and has the 
value of I the average shear on the whole section. If 5 
be that maximum intensity of shear and if 5 is the total 
external transverse shear at the given section, then will 



Art. loo.j SHEARING STRESSES AND WEB REINFORCEMENTS. 621 

5=-7T-. In Fie:, i, Oc = Oa=s and both of the curves 

Ae'a and Bee are parabolas with the vertices at a and c, 
so that horizontal ordinates from AO to the curve in the 
one case and from BO to the curve in the other case repre- 
sent intensities of the longitudinal and transverse shear at 
the points from which those horizontal ordinates are drawn. 
This is the condition of the shearing stresses in beams of 
a single material subjected to flexure, and reinforced con- 
crete beams represent similar members, but of two materi- 
als. The stresses given to the longitudinal steel reinforce- 
ments may be assumed provisionally to be conveyed to them 
from the neutral surface at a constant intensity si and in 
Fig. I that constant intensity is represented by cd and ah. 
The curves df and hf are drawn so as to make a constant 
horizontal ordinate between them and the parabolas already 
indicated. The total maximum intensity of longitudinal or 
transverse shear at the neutral axis will, therefore, be the 
sum of 5 and si\ this may be taken with sufficiently close 
approximation, at least for practical purposes, as | the 
total average transverse shear at a given section.* Even 
if the horizontal ordinate between the two curves is not 
uniform, this value of the maximum intensity may properly 
be used. 

In the case of the tensile resistance of the concrete 



* It has come to be the practice, for some reason not easily appreciated, 
to treat the transverse shear in the normal section of a reinforced concrete 
beam as if it were uniformly distributed over that normal section, which 
is an error on the side of danger. In the interests of both safety and cor- 
rect analysis, the established variation of intensity of shear in the normal 
section of a bent beam should be recognized, for it holds just as much for 
a resisting concrete section as for a section of any other material. When 
the bending resistance of the concrete on the tension side is ignored, the 
law of variation of intensity will change, but the maximum intensity at the 
neutral axis will be unchanged. 



622 



CONCRETE-STEEL MEMBERS. 



[Ch. XIII. 



being neglected, Fig. 2, representing a part of a continuous 
reinforced concrete T-beam, shows the variation of the in- 
tensity of the longitudinal and transverse' shears. The 
parabolic curve Aa shows the variation of the intensity of 
the shear in passing from the neutral axis of the section 
to the exterior surface A, aO being the maximum intensity 
and equal to | the average intensity for the entire section. 
Inasmuch as the tensile resistance of the concrete is neg- 
lected, the maximum intensity of longitudinal shear Ob =0a 
may be considered as varying by some unknown law such, 
however, as to make the total internal transverse shear 




Fig. 2. 



equal to the total external, cd representing the intensity 
at the centre of the tension steel reinforcement. 

It is impossible to analyze with complete accuracy the 
variation of the intensity of shear in the concrete by which 
the reinforcing steel, either in tension or compression, ac- 
quires its stress, but it cannot have a uniform value equal 
to the maximium intensity at the neutral surface. It is to 
be understood that the constant horizontal shear ordinates 
in both Figs, i and 2 are to be interpreted in this 
manner. 

The shearing resistance of concrete in any plain or 
reinforced concrete structure is of uncertain value, much 
as is the tensile resistance, although the practice of crediting 



Art. loo.] SHEARING STRESSES AND WEB REINFORCEMENTS. 623 

it with some material amount may be justified. At the 
same time the incipient surface cracks which are found to 
form with lapse of time at any point may extend deep 
enough ultimately to prejudice seriously resistance to shear. 
It is probably hazardous, therefore, to depend upon concrete 
alone to resist transverse shear in beams, either of the T 
form or solid rectangular, or, of any other form. In fact 
it is prudent to state unqualifiedly that reinforced concrete 
beams carrying moving loads tending to produce vibrations 
or shock should be so designed as to provide for the entire 
transverse shear independently of the shearing resistance of 



, h J c 



M 



F' 



y///// //////A 



C' J' 



N H' 



m:^^^ 



D' K' 



Fig. 3. 



the concrete. This provision for resistance to transverse 
shear is made chiefly by bending upward, in that part of 
the spans near the end supports, the steel tension reinforce- 
ment as shown in Figs. (2) and (3). The inclination of 
the bent parts of the rods will depend upon the judgment 
of the engineer in view of the length of the span, depth of 
beam or other features of each case. Usually all of the 
rods constituting the tension reinforcement are not bent 
upward, as that much provision for shear is not needed. If 
the span is short, there may be but one set of bent rods, 
as shown in Fig. 2, but in other cases there may be two or 
more sets bent upward at different distances from each 
end of the span, as shown in Fig. 3, the number of such 
sets of rods being determined, Hke the angle of inclination, 
in accordance with the best judgment of the designing 
engineer. The vertical components of the stresses in these 



624 CONCRETE-STEEL MEMBERS, [Ch. XIII. 

bent rods may obviously equal the transverse shear at the 
section considered. For example, in Fig. 3, the total trans- 
verse shear at the section CD multiplied by the tangent of 
the inclination of the rods to the vertical must for good 
design be at least equal to the required horizontal rein- 
forcement to be afforded by those rods, i.e., the section of 
the rods must be sufficient to take their stresses without 
exceeding the prescribed working stress, which is frequently 
16,000 pounds per square inch. A similar computation is 
to be made at other points where rods are to be bent up- 
ward. It is not necessary (although usual) that the differ- 
ent sets of inclined parts of rods should be parallel, i.e., 
those nearer the centre of the span may have a greater 
inclination to a vertical than those near the points of 
support. 

Again, vertical reinforcing pieces or stirrups, such as 
those at FH, F'H', CD, C'D\ in Fig. 3, are introduced 
under the assumption that they will take the vertical trans- 
verse shear in tension. These stirrups are of a variety of 
forms and may be in sets of two or more vertical prongs 
or parts, but they should be securely fastened to the hori- 
zontal steel reinforcem^ent. If the total transverse shear at 
any section as JK is supposed to be carried by the stirrup 
as tension in that section, the cross-sectional area of the 
steel stirrups should be sufficient for that purpose at the 
prescribed working stress. Furthermore, the adhesive shear 
or bond on the exterior surfaces of these stirrups should 
be sufficient to give such tension without exceeding the 
prescribed working stress for that shear or bond. These 
vertical stirrups are thus supposed to act the part essen- 
tially of vertical truss members in tension and so produce 
diagonal stresses of compression irf the concrete as shown 
by broken lines in the vicinity of KC and K'C . It is 
known that the greatest diagonal stresses of tension and 



Art. loo.] SHEARING STRESSES AND WEB REINFORCEMENTS. 62S 

compression exist at angles of 45 degrees with the neutral 
surface of every bent beam. The function of these stirrups 
is intended to be such as to relieve the concrete of that 
tension and induce diagonal stresses of compression. Indeed 
their function is somewhat similar to that of vertical stiffen- 
ing members on the web plates of plate girders when those 
stiffeners are assumed to take tension and produce com- 
pression in the web in a 4 5 -degree direction, as was fully 
shown in Art. 34. The distance apart of these vertical 
stirrups should certainly not be greater than the depth of 
the beam from the upper surface down to the tension steel 
reinforcement; probably a horizontal distance apart of 
about three-quarters of that depth is advisable. 

If any beam carry a set of loads, Wi, W2, W3, etc., and 
if R' is the end shear at A, Fig. 3, and if IW be the sum of 
the loads between the end A and any section at which it 
is desired to obtain the transverse shear S\ then will 
that transverse shear at any stirrup, as CD, Fig. 3, be 
S' =R' —^W, and it is assumed that the stirrup will carry 
that shear as tension. If f is the allowed tensile stress in 

the stirrup, the sectional area As of the latter will be As =—j-. 

If the intensity of permitted bond shear is s' and if the cir- 
cumference of a stirrup section is 0, and if V is the imbedded 
length of one complete stirrup, including all prongs, then 
must s'oV be at least equal to 5^ Evidently a form of 
cross-section like an oblong rectangle will give much more 
area for bond shear, for a given sectional area, than such a 
section as a circle or a square and it will have a correspond- 
ing advantage for this purpose. 

The ends of all stirrup bars as well as all reinforcing rods 
should be turned or bent at right angles so as to prevent 
slipping at and near the ends. Furthermore, they should 
preferably be looped at top and bottom, around the rein- 



626 



CONCRETE-STEEL MEMBERS. 



[Ch. XIII. 



forcing rods where they exist, so as to bear directly on the 
concrete supplementary to the bond shear. 

Obviously if both the inclined bent rods shown in Figs. 
2 and 3 and the vertical stirrups shown in Fig. 3 effectively 
perform their functions, both would not be needed at the 
same part of a beam, but as the effectiveness of each detail 
by itself is somewhat uncertain, both are frequently used 
concurrently. The stirrups may judiciously be used in the 
central part of the span extending well toward the ends 
where the bent rods are employed. 

Another form of steel reinforcement of beams is shown in 
Fig. 4, which is supposed to be part of a reinforced con- 




crete beam on both sides of CD, the centre of span. The 
beam may be either a T-beam or a beam of rectangular 
section. The steel reinforcement AB is supposed to be on 
the tension side of the beam only, although a precisely 
similar reinforcement might be placed on the compression 
side also. The small inclined bars ab, cd, a'h' , c'd', etc., 
are usually parts of the main tension reinforcement bent 
upward in a diagonal direction, as shown, which may be 
at the angle of 45 degrees of theory or at some other angle. 
They should extend above the neutral surface NS and be 
carried nearly to the top of the beam. 

As has already been indicated, a solid beam of a single 
material will have the greatest intensity of tensile stress at 



Art. loo.] SHEARING STRESSES AND WEB REINFORCEMENTS. 627 

the neutral surface, making an angle of 45 degrees with a 
horizontal line and sloping upward and away from the 
centre of span, as indicated in Fig. 4. Tests of plain and 
reinforced concrete beams show that in those parts of the 
span near the end, this diagonal tension is likely to cause 
failure of the concrete. Hence these inclined bars are run 
up from the main tension steel reinforcement to assist the 
concrete in taking up this diagonal tension and thus pre- 
venting its failure so far as possible. The concrete will be 
in compression in the diagonal direction at right angles to 
these inclined bars. 

If a vertical section of a beam should cut two or more 
sets of them, the force or stress obtained by multiplying 
half the total transverse shear at such a section by the 
secant of the inclination of these bars to a vertical line will 
give the total stress to which those two or more sets will 
be subjected, the distribution being assumed to be uniform 
among them. The other half of the transverse shear at that 
section may be considered as giving compression to the 
concrete at right angles to the 4 5 -degree tension in the 
inclined bars. It is clear also that the total bond shear 
on the surface of each one of the inclined bars must be at 
least equal to the tensile stress which the bar carries at an 
intensity not greater than that prescribed in the speci- 
fications for the work. If such a normal or vertical sec- 
tion of the beam cuts but one set of these inclined bars, 
the single set must take the stress due to half the total 
transverse shear, precisely as described above for two or 
more sets. The ends of these inclined bars should be bent 
at right angles or otherwise formed so as to prevent the 
possibility of slipping, and thus supplement effectively the 
bond shear. 

It is clear that such bars must add to the carrying 
capacity of a beam, not only by taking up the inclined 



628 CONCRETE-STEEL MEMBERS. [Ch. XIII 

tensile stresses as described, but also as tending to bind the 
entire beam together as a unit. 

The greatest transverse shear is that at the ends of 
the span where the surn of the vertical components of the 
stresses in the bent rods must be equal to that end shear or 
to so much of it as may be prescribed. In Fig. 3, for 
example, the sum of the vertical components of the stresses 
in the inclined rods bK (a set of reinforcing rods) must be 
equal to the transverse shear prescribed. All inclined rods 
like bK, JD, etc., lie in the direction of the diagonal tension 
(maximum at inclination of 45 degrees) and act directly 
to carry shear. 

When beams are continuous over supports, as shown in 
Figs. 2 and 3, bending moments will be developed over those 
supports opposite in sign to those found at and near the 
centres of the spans, producing tension in the upper parts 
of the beams. For this reason tensile steel reinforcement 
formed either of the bent rods continued into the adjoining 
spans, as shown in Fig. 3, or of separate rods introduced 
for the purpose are required to take that tension. 

The precise degree of constraint when beams or girders 
are " continuous " over points of support cannot be deter- 
mined, but certain values of moments expressing the results 
of experience in modifications of formulae for conditions of 
perfect continuity will be given in the next article. 

In the practical consideration of provision for transverse 
shear in reinforced concrete beams, it is a matter of some 
uncertainty how much the concrete may be allowed to take, 
if any, and hence what corresponding steel must be intro- 
duced in the form of bent reinforcing rods or stirrups. As 
has been intimated, it is a serious question whether the 
concrete should be credited with any resistance to shear. It 
is frequently the practice to assume that one-third of the 
transverse shear will be carried by the concrete under suit- 



Art. loi.] STRESSES IN REINFORCED CONCRETE DESIGN. 629 

able conditions and a prescribed working stress, but that 
the other two-thirds shall be taken by steel provided for 
the purpose as already described. Aside from the difficulty 
arising in the attempt to distribute by measure the dis- 
charge of" an important function between two different 
methods, there is grave doubt about the propriety of 
assuming dependable resistance against shear by con- 
crete, particularly if the moving load is of a character to 
produce vibrations or shock. In the latter case steel should 
certainly be provided to take all shear. That procedure 
is more prudent in all cases except, possibly, where the load 
is wholly dead or essentially so, when the concrete may be 
allowed to carry one-third of the total transverse shear. 

Many tests of full-size reinforced concrete T-beams and 
beams of rectangular section have been made by Profs. 
Talbot, Withey, Hatt, and others in the United States and 
by Considere, Feret and other foreign investigators in 
Europe, and full descriptions of all results may be found in 
the Bulletins of the Universities of Illinois and Wisconsin 
and in many other publications, hence it would be super- 
fluous to repeat them here. The working results of those 
tests bearing upon computations for design or other prac- 
tical work will be given in the next article, chiefly in con- 
nection with the recommendations of the " Report of the 
Committee on Concrete and Reinforced Concrete " of the 
American Society for Testing Materials, Vol. XIII, 1913. 

Art. loi. — Working Stresses and Other Conditions in Reinforced 
Concrete Design.* — Design of T-beams. 

In the design of reinforced concrete beams there are 
some features of the work determined by experience and 

*The report of the Committee on Concrete and Reinforced Concrete, 
Proc. of Am. Soc. for Testing Materials, Vol. XIII, 1913, has largely been 
used in the preparation of this article. 



630 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

quite independent of analysis. Reinforcing bars or rods 
must be surrounded by enough concrete to receive the 
proper stress from the latter. This may be assumed to be 
done if the lateral spacing between the centres of parallel 
rods is not less than three diameters, or two diameters from 
the outer concrete surface to the centre of the nearest rod, 
the clear vertical space between two horizontal layers of 
rods being not less than i inch. It is seldom advisable 
to use more than two courses of such rods. In all cases 
scrupulous and effective care should be taken by the aid 
of blocking, ties and other devices to hold the reinforcing 
steel accurately in place until the concrete is set. 

As a fire protection a thickness of at least 2 inches 
of concrete should be placed outside of the steel in all rein- 
forced concrete beams and columns. In relatively small 
beams a least thickness of ij inches may be allowed, and 
I inch may be permitted in floor slabs. 

Floor slabs should be designed and reinforced as con- 
tinuous over supports, and if the length in any case exceeds 
1.5 the width transverse reinforcement should be provided 
to carry the entire load. 

The continuity of beams and slabs may be recognized 
and expressed as follows, assuming the combined dead and 
moving loads equivalent to a uniform load of q (pounds) 
per linear foot on the effective span : 

Floor slabs: moment at centre of span and at 

end of span ^— . 

12 

Beams: For exterior span of series, moment at 

qP 
centre of span and at outer fixed end of span.^— . 

10 

For interior spans moment at centre 

qP 

and at end of span ^— . 

12 



Art. loi.] STRESSES IN REINFORCED CONCRETE DESIGN. 

■Beams and Slabs: continuous over two spans 
only, moment at central support 

Moment near middle of span 



631 



qP 

8 ■ 

qP 

10' 



At ends of continuous beams and girders where the 

degree of constraint is uncertain, the computation of the 

negative end bending moment must be controlled by 
the judgment of the responsible engineer. 

Working Stresses. 

The following working stresses are chiefly given as per 
cents, of the accompanying ultimate compressive resistances. 
They are for moving and dead loads considered as static 
loads, with the assumption that proper additions to moving 
loads must be made, when advisable, to provide for impact 
or vibrations. 

ULTIMATE COMPRESSIVE RESISTANCES 



Aggregate. 


Proportions and Ult. Resistances, Pounds per 
Sq. In. 


1:1:2 


I : 1^:3 


1:2:4 


I : 2^:5 


1:3:6 


Granite, trap rock 


3.300 

3,000 

2,200 

800 


2,800 

2,500 

1,800 

700 


2,200 

2,000 

1,500 

600 


1,800 

1,600 

1,200 

500 


1,400 

1,300 

1,000 

400 


Gravel, hard limestone, 

sandstone 

Soft limestone and sandstone 
Cinders 



Working Compression in Extreme Fibre of Beam. 

The working intensity in the extreme compression fibre 
of a beam may be taken at 32.5 per cent, of the ultimate 
compressive resistance as determined by testing concrete 
cylinders 8 inches in diameter and 16 inches high at the age 



632 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

of 28 days. If, for instance, the ultimate compressive resist- 
ance of a I : 2 : 4 concrete is 2200 pounds per square inch, 
then the extreme fibre working stress would be 2200X.325 
= 715 pounds per square inch. Adjacent to the support of 
continuous beams, these stresses may be increased 15 per 
cent. 

Shear and Diagonal Tension. 

For beams with horizontal reinforcing bars only, i.e., 
with no web reinforcement, 2 per cent, of the ultimate com- 
pressive resistance may be allowed. If the latter were 2200 
pounds per square inch, as for the 1:2:4 concrete of the 
above table, the allowed shear would be .02X2200=44 
pounds square inch. This shear would be taken wholly by 
the concrete. 

For beams thoroughly reinforced in the web, 6 per cent, 
of the ultimate compressive resistance rnay be allowed. In 
this case, however, the web reinforcement, exclusive of bent- 
up reinforcing bars, must be designed to take two-thirds 
of the external vertical shear. Again, using the 1:2:4 
concrete, the allowed shear would be 0.06X2200 = 132 
pounds per square inch of total concrete section. In this 
case, however, the steel reinforcement would be designed 
to carry two-thirds of the total transverse shear, making 
the actual shear in the concrete 44 pounds per square inch 
on the basis of the exact division between the two methods 
of carrying the shear prescribed. 

" For beams in which part of the longitudinal reinforce- 
ment is used in the form of bent-up bars distributed over a 
portion of the beam in a way covering the requirements for 
this type of web reinforcement, the limit of maximum ver- 
tical shearing stress " may be taken 3 per cent, of the ulti- 
mate compressive resistance. 

Where what is termed "punching shear" occurs, i.e., 



Art. loi.] STRESSES IN REINFORCED CONCRETE DESIGN. 633 

]3ure shear without bending, a working shearing stress of 
6 per cent, of the ultimate compressive resistance may be 
allowed. 

Bond or Adhesive Shear. 

The working intensity for this bond or shear between 
concrete and plain reinforcing rods may be taken at 4 per 
cent, of the compressive resistance, but 2 per cent, only for 
drawn wire. For i : 2| : 5 concrete at 1600 pounds per 
square inch of ultimate compressive resistance, the two 
working stresses would be .04 X 1600 =64 pounds per square 
inch or half that for drawn wire. 

Steel Reinforcement. 

The tensile or compressive working stress in steel rein- 
forcement should not exceed 16,000 pounds per square inch. 

Modulus of Elasticity. 

For computations locating the neutral axis and for com- 
puting the resisting moment of beams and for compression 
of concrete in columns, it is recommended that the ratio 
of the steel modulus divided by the concrete modulus be 
taken at 15 if the ultimate compressive resistance of the 
concrete is taken at 2200 pounds per square inch or less; 
and at 12 if the ultimate compressive resistance of the con- 
crete is greater than 2200 pounds per square inch and less 
than 2900 pounds per square inch; and, finally, at 10 if the 
ultimate compressive resistance of the concrete is taken 
greater than 2900 pounds per square inch. 

The preceding specifications express substantially the 
views of a Committee on Concrete and Reinforced Concrete 
of the American Society for Testing Materials, 19 13. In 
that Report the transverse shear is computed as ifuni- 



634 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

formly distributed throughout the normal section of a bent 
beam, which, as has already been indicated, is incorrect. 
On the whole, however, the recommended values are judi- 
cious and may be commended for practical use. 

Design of T-Beam for Heavy Uniform Load. 

The given data are as follows: Effective length of span 
32 feet (non-continuous); moving load on floor, 250 pounds 
per square foot. Floor slab 6 inches thick. Each T-beam 
carries 10 feet width of floor. 

The steel reinforcement of the beam is on the tension 
side only. 

As the floor slab will be reinforced (at right -angles to 
the beam) its weight will be taken at 155 pounds per cubic 
foot. The concrete will be considered a i : 2 : 4 mixture 
weighing about 150 pounds per cubic foot. 

The floor slab being 6 inches thick, a little less than 
four times its thickness will be assumed as effective com- 
pression flange area on each side of the stem or web of the 
beam. Referring to Fig. i of Art. 96, the following dimen- 
sional data will be assumed for trial computations : 

h' =60 inches; / = 6 inches; 6 = 15 inches. hi=2g 
inches; thickness of concrete outside of steel, 2 inches. 
Trial value of steel ratio or per cent., p = .^ per cent. = .008. 

The working stresses are : 

Compression for concrete: ki = 650 lbs. per sq.in. 
Tension for steel: / = 16,000 lbs. per sq.in. ■ 

e = is. 



Eq. (13) of Art. 96 then gives: 

-1.nzt1.524 = +.4i4.-.(ii =.414/^1 = 12 ms. 



di 



hi 



I 



Art. loi.] DESIGN OF T-BEAM. 635 

Eq. (18) of Art. 96 may now be used: 

p = X .414 = .0084 = .84 per cent. 

2 X 16,000 

This last value of p agrees closely enough with the 
assumed value. Hence the computed values of di and p 
may be accepted. Consequently: 6/3 = 29 — 12=17 inches. 

The required steel sectional area is : 

^2 =:/?&'/ii =.0084X60 X 29 = 14.62 square inches. 

There may then be taken eight ij-inch round bars 
whose aggregate area is 14.16 square inches. 

The cross-section of the effective beam may be made 
as shown in Fig. i. Deformed rods of any suitable section 
of the aggregate computed area may obviously be used. 

The dead load or own weight of the beam, including 
10 feet width of floor slab, may be taken at 1225 pounds 
per linear foot of span. The uniformly distributed moving 
load will be 10 X 2 50 = 2500 pounds per linear foot of span. 
The bending moment produced by these two loads will be: 

M = (25oo + i225)^ — -^Xi2 =5,028,800 inch pounds. 


The resisting moment of the beam section must now 
be computed by the aid of eq. (8), Art. 97. 

-^ = 1-25; ^ = 4; di-f = 6; —=.944; ds = i7; di = i2 

and ^ = 15. 

Introducing these numerical quantities in eq. (8), Art. 97 : 

M = 5,024,611 inch-pounds. 
This result is substantially equal to the external bending 



636 



CONCRETE-STEEL MEMBERS. 



[Ch. XIII. 



moment found above and the tentative design may be 
accepted as satisfactory. 

There still remains to be considered suitable provision 
for end and intermediate shears which will be made by 
bending upward the proper number of reinforcing rods 
supplemented by stirrups. 

Fig. I shows to scale about 12 J feet of the T-beam, the 
effective cross-section, 60 inches wide, being shown in shaded 
outline. The dimensions are self-explanatory in connection 
with the computations already made. NS is the neutral 



^T 



j^T P] ' " .^, -,.,.,-, 



*-l>^ 



7777777m- 



=^ 



m^^H. 



ttriril:--: 
L-Tta xirrL- 






ii.- IM 



Fig. I. 



axis. The eight i|-inch round rods in two courses with 
their central line 4 inches from the bottom surface are shown 
both in section and in longitudinal broken lines. This 
latter dimension allows a fire-protecting shell of concrete 
2 inches thick and i inch clear vertical distance between 
the two layers of four rods each. 

The combined dead and moving load on the beam has 
already been shown to be 3725 pounds per linear foot, 
making the end shear 3725X15=55,875 pounds. If bent 
rods inclined at an angle of 45° be supposed to take 
this whole shear, the total stress in those rods will be 
55,875 Xsec. 45 degrees = 79,007 pounds. If the steel be 
stressed at 16,000 pounds per square inch, a little less than 



Art. loi.] DESIGN OF T-BEAM. 637 

5 square inches of section will be required. Three i^-inch 
rounds, or their equivalent sectional area, will supply the 
desired section. It will be convenient to bend the upper 
set of four rods as shown in Fig. i , thus reducing tho actual 
stress in the inclined parts to about 12,000 pounds per 
square inch, the reduced unit stress not being objectionable. 
A greater vertical depth of concrete would have been avail- 
able for shear if the lower set had been bent upward, 
but with the use of stirrups this is not important arrd the 
arrangement shown is a little more convenient in actual con- 
struction. If desired the lower set could be bent, but it 
would be necessary to slightly rearrange the position of all 
the rods so that the bent parts of the lower set may pass 
the upper set, all of which is quite feasible. The hori- 
zontal ends of the bent rods should also be bent at right 
angles so as to secure the firmest possible hold on the con- 
crete at the end of the beam. The horizontal ends of the 
bent bars are about 12 inches long, making the lower bend 
of the same rods about 3.25 feet from the end of the beam. 

Vertical stirrups, 24 inches apart, will be placed through- 
out the central part of the beam and they will be carried 
down so as to pass under the lower reinforcing rods. There 
will be four prongs to each stirrup, looped at top and bottom. 
By this arrangement of the stirrups the bond shear on their 
surfaces is greatly reinforced by the vertical bearing on the 
concrete and reinforcing rods at the bottom. The first 
stirrup, as shown, will be placed at the lower bend in the 
upper set of reinforcing rods, although the stress in it is 
indeterminate, as the inclined rod is supposed to take the 
total shear. 

The total transverse shear in the second stirrup, 5.25 feet 
from the end of the beam, will be computed as carrying in 
tension 9.75X3725=26,320 pounds, requiring at 16,000 
pounds per square inch, 2.25 square inches. Four i|-inch X 



638 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

|-inch flat bars will give the required area, each such flat 
bar constituting one member or prong of the stirrup. The 
shear at the next stirrup point, 2 feet farther from the end 
of the span, will be 28,870 pounds, and four i^-inchX^- 
inch stirrup sections will give a little more than needed, and 
that jection of bar will be adopted. Although smaller bars 
would be sufficient for the remaining sections, the i|-inch 
X A-ii^ch bars will be retained for the remaining stirrups. 

The total available concrete section for resisting shear 
is 29 inches X 15 inches =435 square inches which, under the 
specifications of the preceding article, may be taken at 44 
pounds per square inch, making a total shear of 19,140 
pounds to be provided for in this way if it should be con- 
sidered permissible. If the latter procedure were followed 
it would leave but two-thirds of the total transverse shear 
at each stirrup section to be resisted by the steel stirrups. 
In the case of such a heavy beam, however, it is believed 
to be the better practice to take care of all the shear by 
steel reinforcement. 

If 4 5 -degree steel reinforcements attached to the main 
reinforcing rods were used, the length of such inclined bars 
would be about 27 Xsec. 45 degrees =38 inches. Inasmuch 
as half the transverse shear at any section may be assumed 
to produce 4 5 -degree compression at right angles to such 
inclined tension bars, the latter may be computed as being 
stressed by half the transverse shear multiplied by sec. 45 
degrees. The 4 5 -degree tension bars near the end of the 
span under such an assumption would take about 28,000 
pounds only and if there were four of them, each i^ inchX 
i^ inch, they would be sufficient. At intermediate posi- 
tions further removed from the ends, a correspondingly 
smaller section might be used. The bond shear at the sur- 
face of such inclined bars could be taken at a working 
stress of 88 pounds per square inch of surface. Such in- 



Art. loi.] DESIGN OF T-BEAM. 639 

clined tension bars should be placed not more than about 
21 inches apart horizontally in order to secure effective 
action. Their upper ends should be bent at right angles or 
looped to secure a firmer hold on the concrete. 

These computations illustrate clearly the simple pro- 
cedures required in the design of a reinforced concrete 
T-beam. If the beam is of rectangular section, the pro- 
cedures are precisely the same, as the actual rectangular 
section in that case would correspond precisely to the effect- 
ive shear section taken for the T-beam. 

Design of Continuous Floor Slab for 6 Feet Span between 

Steel Beams. 

The slab is assumed to carry a warehouse load of 175 

pounds per square foot in addition to own weight. It 

will also be assumed to be continuous over the steel beams 

6 feet apart centres, the degree of continuity being that 

prescribed in Art. 100, making the centre and end bending 

wl^ 
moments each — , w being the load per Hneal foot of span. 
12 

A trial depth of slab of 4 inches will be assumed and the 

design will be made for a 12 -inch width of slab. A depth 

of I inch of concrete will be taken outside of the steel 

reinforcement, which will be wholly on the tension side of 

the slab, and the tensile resistance of the concrete will be 

neglected. The data to be used will then be: 

Span =6 feet. Moving load = 175 pounds per square foot. 
Dead l^ad = 50 pounds per square foot. 

Tension in steel, t = 16,000 pounds per square inch. 
Compression in concrete, ki =500 pounds per square inch. 

-=r = e=is; /i =4 inches; /^i =2.75 inches; 6 = 12 inches. 



640 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

22 c X 6 X 6 

The external bending moment, M = — ^ X 12 =8100 

12 

inch-pounds. The section to be designed must give a 

resisting moment at least equal to 8100 inch-pounds. 

Eq. (8), Art. 98, gives the steel ratio: 

;p = .005 = . 5 per cent. 
Hence, 

^2 = .005 X4 X 12 =.24 square inch. 

Eq. (4), Art. 98, then gives the position of the neutral 
axis: 

di 

T-= -.o75±.394=+.3i9; 
hi 

and 

di =0.88 inch; 

ds =hi —d\ = 1.87 inches. 

The internal resisting moment will now be given 
eq. (12), Art. 98: 



500/12 X.^^ 



. M=^N^^-^^^ -f3.6X1.87' =8700 inch-pounds. 
.88 \ 3 / 

By revising the design the excess above 8100 inch-pounds 
may be reduced if desired, but the difference is too small 
to be material. 

Two f-inch square bars, placed 6 inches apart, having 
a combined area of .28 square inch, will afford satisfactory 
reinforcement, remembering that they must be carried 
from I J inches above the lower surface of the slab at the 
centre of span to that distance below the upper surface 
at the ends of the span. 

The end shear of 3X225=675 pounds is provided for 



Art. I02.] REINFORCED CONCRETE COLUMNS. 641 

by the bending up of the reinforcing rods, especially as 
the concrete section is 4X12 =48 square inches. 



Art. 102. — Reinforced Concrete Columns. 

Reinforced concrete columns may be divided into 
two classes. The reinforcing steel in one of these classes 
is a wrapping or banding, usually as a spiral, of the concrete 
by coarse wire or thin fiat bars, so that the lateral strains 
or enlargement due to axial compression will be prevented 
as much as possible with the intent to increase correspond- 
ingly the carrying capacity of the col- 
umn. It is customary to use longi- 1^ ^ ^ 

tudinal steel rods spaced equidistantly ! ^^ -^ ] 

around the column adjacent to and //^^ ^M 
inside of the spiral banding, as shown i/px p\V^ 

in Fig. 2, the former being strongly 11 jj 

fastened to the latter by clamps or \\ // 

wires. When the cylindrical cage ^^^TZT^^^^ 

thus formed is filled with concrete, ^ig i 

usually a rich mixture such as i : 2 14, 

and encased with concrete about 2 inches thick, the com- 
plete column is formed. 

The steel reinforcement in the other class of columns 
is a load-carrying member, in fact a steel column in itself, 
filled with concrete and encased with the same exterior 
shell of concrete as in the banded column, as shown in 
Fig. 3. In the latter case the parts of the steel column 
reinforcement form the banding or wrapping around the 
concrete. The shape of cross-section of column for either 
class may be any desired, although the circular section 
is more convenient for the first class. 



642 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

Lateral Reinforcement and Shrinkage 

The analytic expression for the gain in carrying 
capacity arising from banding is easily written. Let Fig. 
I represent a band one unit (inch) in length, i.e., along the 
axis of the column, its interior diameter being d. When 
the column receives load its diameter d tends to increase 
in consequence of the lateral strains, thus pressing against 
the interior of the band and causing the latter to stretch 
accordingly. Let 

£^2 =30,000,000 = modulus of elasticity of the steel; 
El = 2,000,000 = modulus of elasticity of the concrete; 
px = uniform intensity of pressure between the ring 

or band and concrete ; 
pi = intensity of column loading on a normal section; 
a = area of section of band ; 
A = stretch of steel ring due to internal pressure px- 



Hence 



A =~-Trd. .*. New circumference =7r(i( i +^) . (i) 
ii2 \ A2/ 



The new diameter will be <i( i +-^ j . 

If r is the ratio between the direct compressive and lateral 
strains for concrete, the new diameter of the banded con- 
crete will be: 



Equating the two values of the new diameter, 
£2 El El El +£2 



P'_P^r_p. . _ ^^_Pj,. . . . (3) 



Art. 102.] REINFORCED CONCRETE COLUMNS. 643 

Eq. (3) gives the value of the intensity of pressure be- 
tween the banding and the concrete. If ^r- = i5, 

^x = J|^ir ....... (4) 



If for concrete r =-, 

5 

P^ = f^P^ . (5) 

With this value of r, p'x=pir =— would prevent all 

lateral strain, and as eq. (4) shows that px = -^p'x, it is clear 

16 

that the banding appears to be highly effective. Con- 
crete, however, shrinks when it sets in air with a coefficient 
of shrinkage, according to such tests as have been made, 
of .0002 to .0005. If £1=2,000,000 and if, for example, 
pi = 1500 pounds per square inch, then by eq. (5), 



^Xisoo 

= — — (nearly) (6) 

2,000,000 7000 



As both and are greater than -, these 

5000 2000 7000 

computations show that shrinkage of concrete setting in air 
will more than neutralize the advantage supposed to be 
due to banding, at least until the elastic limit of the con- 
crete, and probably the yield point, is exceeded. This 
explains why banding shows no advantage in full-size 
column tests until the yield point is passed, as will be seen 
later. 



644 CONCRETE-STEEL MEMBERS. [Ch. XIII. 



Longitudinal Reinforcement 

In considering the effect of longitudinal steel reinforce- 
ment, let 

A = total available sectional area of the column (the 
outer 2 -inch thickness is neglected in computa- 
tions for carrying capacity) ; 
A 2 = sectional area of steel; 
A 1 = sectional area of concrete ; 

p = steel ratio —^ ; 

pi = intensity of compression in concrete; 
c = intensity of compression in longitudinal steel; 

A=Ai+A2 and e=^. 

El 

P = carrying capacity of reinforced column ; 

^1= carrying capacity of plain concrete column of 

section A. 

Hence: 

P=cA2+pi{A-A2)={pei-i-p)piA. . (7) 



Or 



pr=p(e-i)+i (8) 



Eq. (8) shows the gain of carrying capacity due to the 
longitudinal reinforcing steel. The fractional gain is: 

^-ph=Pie-i) (9) 

Many tests of full-size columns of both types have been 
made by Professors Talbot, Withey, the author, and others, 
the results of which fully described may be found in the 
bulletins of the Universities of Illinois and Wisconsin, 



Art. I02.] REINFORCED CONCRETE COLUMNS, 645 

and those of the author in the " Proceedings of the Insti- 
tution of Civil Engineers " of London. 

The effect of a proper amount of spiral or other band- 
ing, either by itself or in connection with longitudinal 
steel rods firmly secured to it, or of a self-supporting load- 
carrying steel column, is, in all these types, to support the 
concrete to such a degree as to develop substantially its 
ultimate carrying power in short blocks, for such lengths 
of columns as have been tested. 

In order to accomplish this result i per cent, of 
lateral steel reinforcement in spiral shape is sufficient. 
Furthermore it is preferable to use longitudinal steel rod 
reinforcement in connection with the lateral spiral rein- 
forcement, the two being firmly attached to each other 
in all cases. The spiral cage firmly secured to the longi- 
tudinal rods constitutes practically an independent load- 
carrying steel column, particularly when filled with con- 
crete. A properly designed reinforced column of this type 
may have its ultimate carrying capacity closely represented 
by eq. (7), in which pi is the ultimate compressive resist- 
ance of the concrete and c the ultimate compressive resist- 
ance of the steel. If longitudinal rods are used without 
the steel banding, they cripple or buckle under compara- 
tivel}^ light loads, as would be expected, and make an 
unsatisfactory column in combination with the concrete 
of reduced carrying power. 

As has already been shown in connection with eqs. (i) to 
(5) the shrinkage of the concrete in setting prevents the 
banding influence of the steel from being effective until 
the yield point of the concrete has been passed, and the 
results of tests have confirmed fully the indications of 
analysis. The same tests, however, have shown that in 
properly designed columns of both classes the concrete 
and the steel act together effectively except in the case of 



646 



CONCRETE.STEEL MEMBERS, 



[Ch. XIII. 



longitudinal rods without spiral or other banding. This 
latter type of column, however, is too indifferent in char- 
acter to be used in practice. 

Types 0} Columns, 

Figs. 2 and 3 illustrate the two types or classes of 
reinforced concrete columns generally used. Fig. 2 shows 






Fig. 2. 



Fig. 3. 



a spiral reinforcement inside of which there are a suitable 
number of longitudinal round or other rods which must 



Art. 102.] REINFORCED-CONCRETE COLUMNS. 647 

be firmly secured to the spiral reinforcement. The size 
of the latter may vary according to the size of the column 
from J inch diameter to f inch or more, and the pitch may 
vary from i to several inches, according to the size of the 
column. . It has been found, as already indicated, that the 
amount of spiral or lateral reinforcement should be about 
I per cent., i.e., the volume of the spiral metal should be 
about I per cent, of the volume of the column, counting the 
diameter of the latter as the diameter of the cylinder 
formed by the centre line of the spiral. The amount 
of longitudinal steel rods may be i| to 2 or 3 per cent, 
or more; although it has been found generally to be more 
economical to increase the richness of the concrete core 
and use less longitudinal steel than to use more of the latter 
with leaner and less expensive concrete. The exterior 
concrete shell, usually about 2 inches thick, is not con- 
sidered as an available or load-carrying part of the com- 
plete column. It has been found by experiment that this 
exterior shell may carry from 40 to 50 per cent, as much 
load per square inch as the concrete core, and that it will 
not crack off under test until the yield point of the steel 
has been reached, but it is quite likely to be at least par- 
tially destroyed in a burning building. On the whole, 
therefore, it is considered better practice, and it is certainly 
safer to consider the core only of reinforced columns, i.e., 
only that part within the exterior enveloping volume of 
the steel as load carrying. 

Fig. 3 is typical of the class of columns in which the 
steel is designedly a load-carrying member. The figure 
shows a column of four angles latticed in the usual manner 
with batten plates as well as lattice bars on all four sides, 
but a great variety of forms may obviously be used in this 
type of column. Many full-size tests have shown that the 
concrete filling of this type of column, no less than in the 



64S CONCRETE-STEEL MEMBERS. [Ch. XIII. 

other type, may be considered as carrying load up to its 
full ultimate short block capacity before failure of the 
column for all lengths used in actual tests. The load 
carried by such a column may be computed by adding the 
carrying capacity of the concrete filling considered as a 
short block to the carrying capacity of the steel column 
computed as such. 

Prof. Withey has concisely expressed the results of 
the tests of full-size columns of both these types in the 
Bulletin of the University of Wisconsin as follows : 

" I. A small amount, 0.5 to i per cent., of closely spaced 
lateral reinforcement, such as the spirals used, will greatly 
increase the toughness and ultimate strength of a concrete 
column, but does not materially affect the yield point. 
More than i per cent, of lateral reinforcement does not 
appear to be necessary. The use of lateral reinforcement 
alone does not seem advisable. 

''2. Vertical steel in combination with such lateral 
reinforcement raises the yield point and ultimate strength 
of the column and increases its stiffness. Columns rein- 
forced with vertical steel only are brittle, and fail suddenly 
when the yield point of the steel is reached, but are con- 
siderably stronger than plain columns made from the same 
grade of concrete. 

"3. Increasing the amount of cement in a spirally 
reinforced column increases the strength and stiffness of 
the column. A column made of rich concrete or mortar 
and containing small percentages of longitudinal and 
lateral reinforcement, is without doubt fully as stiff' and 
strong and more economical than one made from a leaner 
mix reinforced with considerably more steel. In these 
tests, doubling the amount of cement increased the yield 
point and ultimate strength of the columns without vertical 
steel' about 100 per cent., and added about 50 per cent, to 



Art. I02.] REINFORCED CONCRETE COLUMNS. 649 

the strength of those reinforced with 6.1 per cent, vertical 
steel. 

"4. From the behavior under test of the columns 
reinforced with spirals and vertical steel and the results 
computed, it would seem that a static load equal to from 
35 ,to 40 per cent, of the yield point would be a safe working 
load. 

** 5. The results obtained from tests of columns rein- 
forced with structural steel indicate that such columns 
have considerable strength and toughness, and that the 
steel and concrete core act in unison up to the yield point 
of the former. The shell concrete will remain intact until 
the yield point of the steel is reached, but no allowance 
should be made for its strength or stiffness." 



"2. Although the yield point of a reinforced concrete 
column is practically independent of the percentage of spiral 
reinforcement, the ultimate strength and the toughness are 
directly affected by it. . . . Consequently, only enough 
lateral reinforcemicnt is needed to prevent the longitudinal 
rods from bulging outward, and to provide an additional 
factor of safety against an overload by increasing the 
toughness and raising the ultimate strength somewhat 
above the yield point. From these tests i per cent, of a 
closely spaced spiral of high-carbon steel seems to be 
sufficient for this purpose. 

''3. By the addition of longitudinal steel the yield 
point, ultimate strength and stiffness of a spirally rein- 
forced column can be considerably increased. If maximum 
economy in floor space is desired, if a column is so long or 
is so eccentrically loaded that tension exists on a portion 
of the cross-section, or if a large dead load must be sus- 
tained by the column while the concrete is green, a high 



650 CONCRETE-STEEL MEMBERS. [Ch. XIII 

percentage of longutidinal reinforcement may often be 
advantageously employed. Such reinforcement is also a 
valuable safeguard against failure due to flaws in the 
concrete. If the cost of cement is extremely high, it may 
be economical to use a leaner mixture than suggested in 
(i) and considerable longitudinal steel to increase the 
stiffness and strength; columns like those of Series i 
may profitably be used. In general, however, cement is 
a more economical reinforcement than steel. Therefore, 
for ordinary constructions it does not seem advantageous 
to use in combination with a rich concrete more than 2 
or 3 per cent, of longitudinal steel." 

"8. Briefly sum_marizing the foregoing, it seems eco- 
nomical to use for reinforced concrete columns a very rich 
mixture, and advantageous to employ about i per cent, of 
closely spaced high-carbon steel lateral reinforcement com.- 
bined with 2 or 3 per cent of longitudinal reinforcement. 
From the test data presented it seems apparent that such 
columns, centrally loaded, may be subjected to a static 
working stress equal to one-third of the stress at yield 
point." 

Working Stresses 

The results of analysis and of the full-size tests to 
which reference has been made furnish a rational basis 
on which proper working stresses may be based. The 
concrete is so held and supported in both types of column, 
when properly designed, that the working stress in it, may 
be prescribed as if it were a short block. In that class of 
columns in which the steel reinforcement is a steel column 
by itself, the working stress in the latter may be prescribed 
precisely as for any other steel column. Manifestly the 
fraction of the ultimate resistance represented by the work- 
ing stresses for both materials must be the same. Actual 



Art. 102.] REINFORCED CONCRETE COLUMNS. 651 

tests of full-size columns enable the unit working stress 
for the longitudinal steel in the spiral-banded columns to 
be properly prescribed, the steel spiral banding being a 

1 per cent, lateral reinforcement not to be credited as 
carrying any direct load. 

The unit compressive working stress of the longitudinal 
steel reinforcing members in either type of column is taken, 
in the recommendations of the American Society for Testing 
Materials in their Proceedings for 19 13, at 16,000 pounds 
per square inch, it being understood that the length of no 
column shall exceed fifteen times the least width, that 
width not including the protective shell, usually about 

2 inches thick. 

The same ratio of length to least width holding for 
both types of columns, the following compressive working 
stresses are recommended by the Committee on Concrete 
and Reinforced Concrete of the American Society of Civil 
Engineers, 1913, the per cents, stated 'to be applied to the 
ultimiate resistances of the various grades of concrete given 
in Art. loi. 

Structural steel in tension 16,000 lbs per sq.in. 

Per cent, of 
Ult. Com- 
pressive 
Resist. 
Concrete in compression where resisting area is at least twice 

loaded area 32 . 5 

Concrete in plain concrete column or pier centrally loaded, length = 

12 diameters or less 22 . 5 

Concrete in column with i to 4 per cent, longitudinal reinforcement 

only; length of column = 12 diameters or less 22 . 5 

Concrete in column with lateral reinforcement of spirals, etc., at 
least I per cent, of volume of column, clear spacing of spirals or 
hooping, rs to ^ of diameter of encased column, in no case ex- 
ceeding 2 1 inches, the length of laterally unsupported column to 
be not more than 8 diameters of hooped core 27 . 

Concrete in column with i per cent, to 4 per cent, of longitudinal 
bars with spirals, hoops, etc., as specified above column, the 
length of laterally unsupported hooped core, not more than 8 
diameters of core 32 . 625 



652 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

Reinforced columns with longitudinal steel rods, only, 
embedded in the concrete are highly unsatisfactory and 
they should not be used where the failure of the column 
would entail serious consequences. 

The tests of such load-carrying columns for steel rein- 
forcement of reinforced concrete as have been made by 
the author show that their ultimate resistances will be closely 
given for such lengths as have been tested by the simple 
straight-line formula 

P I 

-=43,000-155^. 



In this formula - is the ratio of length of column to 
r 

the radius of gyration of its cross-section about the neutral 

p 

axis and A is the area of cross-section. Hence — is the 

average unit compressive stress over a normal section of 
column. 

This type of column is not limited in use to any ratio 
of length over least diameter, nor is the per cent, of steel 
section restricted. As the steel reinforcement is a perfectly 
designed load-carrying column, it may be treated like any 
other steel column, while the concrete filling is so banded 
and supported by the enclosing steel column that load 
may be imposed upon it as in the case of a short concrete 
block. 

These columns have been used for tall buildings of 
eleven stories or more in height. They are well adapted 
to such a purpose, not only in consequence of the load- 
carrying capacity of the steel, but also on account of the 
facility with which floor beams and girders or other members 
may be detailed to them 



Art. I02.] PROBLEMS FOR CHAPTER XIII. 653 

The spiral or otherwise banded column is not so well 
adapted to structural purposes. The design is such that 
they are available only for comparatively short lengths in 
connection with the prescribed working stresses. They may 
probably be used up to lengths of unsupported core equal 
to twelve times the least diameter under a reduction of 
working stresses to 80 per cent, of those prescribed. 

The two following problems will illustrate the applica- 
tions of the preceding results to actual design work: 

Problem I. 

Design a reinforced-concrete column 13 feet 6 inches 
long, with spiral banding and longitudinal rod reinforce- 
ment to carry a load of 354,000 pounds. 

As the column must not exceed 8 diameters in length, 
the diameter of the spiral banding will be taken as 20 
inches, giving an effective area of 314.2 square inches. 
There will be assumed eight i|-inch longitudinal round 
rods arranged as shown in Fig. 2. The concrete will be 
taken as a 1:2:4 mixture with an ultimate resistance 
at twenty-eight days of 2250 pounds per square inch. 
Hence the working unit stress will be: ;^ = 2250X32.625 =734 
poimds per square inch. The working stress, c, of the steel, 
as has been shown by the specifications of the joint com- 
mittee of the Am. Soc. C.E. and the Am. Soc. for Testing 
Materials, may be taken at 16,000 pounds per square inch. 
Hence the total carrying capacity of the column is: 

Of the steel section 8 X 16,000 =128,000 lbs. 

Of the concrete section. (314.2-8) X734=2 44,75i lbs. 

Total 

352,751 lbs. 

This is sufficiently near 354,000 to be considered satis- 
factory and it will be accepted. It illustrates fully the 



654 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

procedure to be followed in the design of this type of 
column. 

The I per cent of spiral lateral reinforcement may be 
determined as follows : The volume of spiral metal per 
inch of length of column is 0.01X314=3.14 cubic inches. 
If the pitch is 2 inches (one-tenth the diameter) the length 
of one corhplete turn of the spiral will be about 63 inches. 

Hence the sectional area of the spiral rod will be - — = . i 

63 
square inch (nearly), requiring a f-inch round rod. This 
close wrapping or banding by a f-inch spiral with 2 -inch 
pitch must be firmly fastened by coarse wire or clips to 
the eight i|-inch longitudinal round rods. 

Problem II. 

Design a reinforced-concrete column 20 feet long to 
carry a load of 283,000 pounds, the steel reinforcement to 
be a load-carrying column. 

Let the reinforcing column be composed of four 3X3X1^- 
inch steel angles latticed to form a column like Fig. 3. 
The square formed by the angles will be 15 inches on a 
side, i.e., from back to back of angles. The radius of 

gyration r of such a section is 6. 7 inches. Hence - =-^ =36, 

r 6.7 
p 
and eq. (10) gives -^ =37,420 pounds per square inch. If 

working stresses be taken at one-third the ultimate, the 

working stress for steel will be c=— ^ = 12,470 pounds 

3 
per square inch. The sectional area of a t,Xs Xi^-inch 
angle is 2.43 square inches. Hence the effective area 
of the concrete section is 15X15—4X2.43=215.3 square 
inches. The concrete will be assumed to be a 1:2:4 
mixture, for which the ultimate resistance may be taken 



Art. 103.] DIVISION OF LOADING. 6$$ 

at 2250 pounds per square inch, and the working resist- 
ance, 750 pounds per square inch. The total carrying 
capacity of the column will then be: 

Of the steel section 12,470X9.72 =121,240 lbs. 

' Of the concrete section . 750X215.3 =161,460 lbs. 



Total 282,700 lbs. 

This result shows that the design is satisfactory. 

Art. 103. — Division of Loading Between the Concrete and Steel 
Under the Common Theory of Flexure. 

It is occasionally desirable to determine the portion of 
the total loading of either a concrete-steel beam or arch 
carried by the steel and concrete parts of the member. 
In making this determination the formula estabhshed in 
the preceding articles in accordance with the common 
theory of flexure will be employed. It will be convenient 
also for this purpose to represent the intensity of stress in 
the extreme fibre of the steel, whether tension or com- 
pression, by k^, the distance of that extreme fibre from the 
neutral axis of the composite section, established in Art. 97, 
being represented by d2. It will further be supposed that 
the coefficients of elasticity for concrete in tension and 
compression are the same. Eqs. (4) of Arts. 96 and 98, 
representing the resisting moment of the internal stresses 
in a normal section of a composite member, may then be 
written 

_ _ kill , ^2/2 . . 

^=-dr+^ (^) 

Let the total load on the composite beam or arch be 
represented by W, while Wi and W2 represent the portions 



656 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

of VV carried by the steel and concrete respectively. Also 
let q^ and q^ be so taken that ^1=717- and g^^w- The 

remaining notation will be that given in x\rt. .96. 

Since the bending moments in the portions A^^ and A^ 

are proportional to the loads which those portions carry, 

k, k 

remembering that -- and -r are equal to E^tc and E^u re- 

spectively, there may be written, as indicated by eq. (i), 




^1^1 arid a =— 2_ ^^2 

EJ, + E,I, "'''' ^^ VV EJ,-\-E,I, 



and ^2 = 



I, + eI, 



(2) 



Also, if ^=^S 



n . I 

Qi= — \ — ^i^d q^= — -— . . . , , (2) 



Then, since M^=q^M and M.^=q^M, 

^.=^ and ..=?f^. ... (4) 

Eqs. (2), (3), and (4) show the portions of loading 
carried by the two materials and the greatest intensities 
of stresses in their extreme fibres. 

It is sometimes necessary to combine a bending moment 
with the direct compression (or tension) produced by a 
force P acting along or parallel to the axis of a beam or 
arch. Let p^ and p^ represent the intensities of stress 
produced in the two portions A^ and A^ by such a direct 
force. Since equal unit longitudinal strains exist in the 



Art.. 103.] DIVISION OF LOADING. 6'S7 

two materials, the intensities of stress in the portions A^ 
and A^ will be proportional to their coefficients of elastici- 
ties. Hence 



^'=-^ and A— f^A- ..... (5) 



Hence 



PA^+ePA.-P; ■••ft=XTM- • • (^) 



In the case of an elastic arch like those of combined 
concrete and steel, the thrust P is in general exerted along 
the axis of the arch ring but at some distance, /, from it. 
In such a case the bending moment is 

M=Pl; hence M,=q,Pl and M,=q,PL . (7) 

The values of the bending moments are to be placed in 
eq. (4), in order to determine the intensities k^ and k^. 

In determining: the resultant of stress for any section 
of an arch ring, if the conditions under which eqs. (2) were 
written be employed, the thrust on the portion A^^ will be 
q^P, and q^P on .4^, since the thrusts on the two portions 
will be proportional to the loads which they carry. Hence, 
if k^ and k^ again be used to represent the greatest inten- 
sities of stress in the two portions, there may at once be 
written 

/P MdA 



/P 'MdA 

^^-^{a.^-t;) ^'' 

In eqs. (8) and (9), M=Pl. 

If, again, the last members of eqs. (5) and (6) be used 



658 CONCRETE-STEEL MEMBERS. [Ch. XIII. 

in connection with eqs. (2) and (4) the resultant values 
of k^ and k2 will be 

^ . q,Md, I P Md, \ , , 

In the use of all these equations, care must be taken 
to give the proper sign to the bending moment M. 

These equations comprise all that are necessary in order 
to ascertain the distribution of the loading between the 
steel and the concrete, or any other two materials, whether 
the case may be one of pure bending or a combination of 
bending and direct stress. 



CHAPTER XIV. 
ROLLED AND CAST FLANGED BEAMS 

Art. 104. — Flanged Beams in GeneraL 

Rolled flanged beams as produced by steel mills and 
used in building or other construction have already been 
treated in cases of simple bending, using the moment of 
inertia either by itself or as part of the section modulus 
for steel beams where their moments of resistance take 
the usual form, 

M=| . . (i) 

In this equation di is the distance of the extreme fibre 
from the neutral axis in which the intensity of stress k 

exists, and I and — are the moment of inertia and the 
di 

section modulus, respectively, numerical values of which 

for all shapes are given in handbooks. In this treatment 

of rolled or other flanged beams the resistance of the web is 

included, but there are cases when it is permissible to neglect 

the bending resistance of the web or, again, in which the 

bending resistance of the two flanges is treated separately, 

as if the intensity of stress in each is uniform throughout 

the flange section, to which a closely approximate simple 

expression for the bending resistance of the web may or 

may not be added. 

If the bending resistance of the flanges is to be com- 

659 



66o ROLLED AND CAST FLANGED BEAMS. [Ch. XIV. 

puted by itself, it is evident that economy of design requires 
that the two flanges must fail concurrently if the beam 
be loaded to failure. If the ultimate tensile and com- 
pressive resistances of the material are not the same, it 
is equally clear that the two flanges should not be of equal 
section, the area of the flange in which the ultimate resist- 
ance is greater being less than that of the flange in which 
the ultimate resistance is less. This results from the 
fact that the total stress of compression in the compres- 
sion flange must be equal to the total tensile stress in the 
tension flange, the beam being supposed to be horizontal 
and the load vertical. If the bending resistance of the 
web is recognized, the equality of the two total flange 
stresses no longer holds, since the tension and compression 
developed in the web is to be added to the corresponding 
stresses in the flanges in order to make equality. 

Each total flange stress is evidently equal to the flange 
area multiplied by the intensity of assumed uniform stress 
in it. The centre of each flange stress will then be the 
centre of gravity of the section on which it acts. The 
vertical distance d between the centres of gravity or stress 
of the two flanges is called the effective depth of the beam, 
because if it be multiplied by either flange stress the prod- 
uct will be the resisting moment of the stresses acting 
in the section of the beam. In other words the effective 
depth d is the lever arm of the internal couple whose moment 
is equal to the external bending moment. 

Let a be the sectional area of the tension flange and T 
the uniform intensity of stress in it, and let a' and C'be 
the corresponding values for the compression flange, while 
d is the effective depth. Then, since aT =a'C, the moment 
of the internal stresses will be 

M=aTd=a'Cd (2) 



Art. 105.] FLANGED BEAMS JVITH UNEQUAL FLANGES. 



661 



The use of both eqs. (i) and (2) will be illustrated by 
numerous practical applications. 

It is clear from what has preceded that the chief 
function of the flanges is to resist the bending proper, 
while the main function of the web is to resist the trans- 
verse shear. 

The direct stresses of tension and compression in a 
beam with solid rectangular section correspond to, i.e., 
perform the same function as, the flange ^tresses of tension 
and compression in the flange beam, while the web, supposed 
to take shear only, corresponds approximately to the zone 
of material in the vicinity of the neutral surface of the 
solid section in which the direct stresses of tension and 
compression are either zero or nearly zero. 



> 5' 



Art. 105. — Flanged Beams with Unequal Flanges. 

By the common theory of flexure, if the two coefficients 
of elasticity are equal, it has been shown that if C, Fig. i, 

is the centre of gravity of the j^~~^ " 

cross-section, the neutral axis [^ 

of . the section will pass through 
that point. Let it now be sup- 
posed that the lower flange is in a__ 
tension, while the upper is in com- 
pression. Also let T represent 
the ultimate resistance to tension 
in bending, and let C represent the 
same quantity for compression in ' Fig. i. 

bending. Then s'nce intensities vary directly as distances 
from the neutral axis, 



.f3 

A, 

I' 

F 



r 



h 



T 



T 

^1 =^7^=^'^- 



(l) 



662 



ROLLED AND CAST FLANGED BEAMS. [Ch. XIV. 



The ratio between ultimate -ntensities is represented by 
n'. U d=h-{-h^ is the total depth of the beam, and hence 

^d 

If, as an example, for cast iron there be taken 

r T J I ' 

n =— = 0.2, hi =-d. 
C 6 

The relation between h and h^ shown in eq. (2) is en- 
tirely independent of the form of cross-section. But 
according to the principles just given, in order that economy 
of material shall obtain, the cross-section should be so de- 
signed that h and h^ shall represent the distances of the centre 
of gravity from the exterior fibres. 

The analytical expression for the distance of the centre 
of gravity from DF is 

ib'a' + (b-b^)f(d-if) + i{b,-b%' ' 

^1 bd-\-{b-b')f-\-{b^-b')t^ ' ' ' ^^^ 

The meaning of the letters used is fully shown in the 
figure. In order that the beam shall be equally strong in 
the two flanges, the various dimensions of the beam must 
be so designed that 

x^=\. ....... (4) 

It would probably be found far more convenient to cut 
sections out of stiff manila paper and balance them upon 
a knife-edge. 



Art. 105.] FLANGED BEAMS IVITH UNEQUAL FLANGES. 663 

The moment of inertia about the axis AB, thus deter- 
mined, is 



I =\W +hih,^ -{h-h'){h-ty -{hi-h'){hi-hY] . (4a) 

This value is 
now changed^ to 



kl 
This value is to be substituted in the formula M=^-, 

di 



For various beams whose lengths are / and total load W 
the greatest value of AI becomes : 
Cantileve uniformly loaded, 

M= — . 
2 

Cantilever loaded at end, 

M = Wl. ' 

Beam supported a' each end and uniformly loaded, 

'"^ 8 8 ■ 
Beam supported a each end and loaded at centre, 

M= — . 
4 



The last two cases combined, 



?)■ 



Sometimes the resistance of the web 's omitted from 
consideration. In such a case the intensity of stress in 



664 ROLLED AND CAST FLANGED BEAMS. [Ch. XIV. 

each flange is assumed to be uniform and equal to either 
T or C. At the same time the lever-arms of the different 
fibres are taken to be uniform, and equal to h for one flange 
and h^ for the other, h and h^ now representing the vertical 
distances from the neutral axis to the centres of gravity of 
the flanges, while d--^h-\-h^. 

On these assumptions, if / is the area of the upper flange 
and f that of the lower, there will result 

M=fC.h+rT.h, (5) 

But since the case is one of pure flexure, 

fC=f'T (6) 

.: M=fC(h + h;)=fCd^f'Td. ... (7) 
Also, from eq. (6), 



f'-C 



(8) 



Or, tne areas of the flanges are inversely as the ultimate 
resistances. 

Frequently there is no compression flange, the section 
being like that shown in Fig. 2. In such 
case h is equal to h' , or t' is equal to zero; 
hence h =h' in eq. (4a) , but no other change 
1 is to be made in the second member of that 



Fig. 2. equation. Eq. (46) may then be used pre- 

cisely as it stands for the internal resisting 
moment of a beam with the section shown in Fig. 2. 

Prob. I. It is required to design a cast-iron flanged 
beam of 5 feet effective span to carry a load of 1800 pounds 
applied at the centre of span, the section of the beam to 
be like that shown in Fig. 2, i.e., without upper flange. 
The greatest permitted working stress in compression will 



Art. io6.] FLANGED BEAMS IVITH EQUAL FLANGES. 66$ 

be 8000 pounds per square inch, and the total depth of the 
beam is to be taken at 9 inches. 

Referring to eqs. (4a), (46), and Fig. i for the notation, 
the given data and the dimensions to be assumed for trial 
will, be as follows: d = g inches; b=b'=l inch; bi=S 
inches; ^1 = 1 inch; / = 5 feet; and C = 8ooo. The intro- 
duction of these values into eq. (3) will give for the distance 
of the centre of gravity above the bottom surface of the 
beam 

/ji =2.6 inches and h=d— hi =6.4 inches. 

The preceding trial dimensions will make the beam 
weigh about 50 pounds per lineal foot. If all the preced- 
ing values are substituted in eqs. (4a) and (46), remembering 

that M = — , there will be found 
4 

W = i994 — 125 =1869 pounds. 

The trial dimensions, therefore, give the centre-load 
capacity of the beam 69 pounds greater than required, 
which may be considered sufficiently near to show that the 
assumed dimensions are satisfactory. 



Art. 106. — Flanged Beams with Equal Flanges. 

Nearly all the flanged beams used in engineering prac- 
tice are composed of a web and two equal flanges. It has 
already been seen that the ultimate resistances, T and C, 
^ of structural steel and wrought iron to tension and com- 
pression are essentially equal to each other ; the same may 
be said a''so of their coefficients of elasticity for tension 
and compression. These conditions require equal flanges 
for both steel and wrought-iron rolled beams. 



666 



ROLLED AND CAST FLANGED BEAMS. [Ch. XIV. 



I 



I 
I 

H— + . 
I 
I 

I 

I 



— B- 



In Fig. I is represented the normal cross-section of an 
equal -flanged beam. It also approximately represents 
what may be taken as the section of c 

any wrought-iron or steel I beam, the ^ — 

exact forms with the corresponding 

moments of inertia being given in hand- Y 
books. Although the thickness f of the ^ 
flanges of such beams is not uniform, 
such a mean value may be taken as 
will cause the transformed section of 
Fig. I to be of the same area as the 
original section. 

Unless in exceptional cases where 
local circumstances compel otherwise, 
the beam is always placed with the web vertical, since the 
resistance to bending is much greater in that position. 
The neutral axis HB will then be at half the depth of the 
beam. Taking the dimensions as shown in Fig. i, the mo- 
ment of inertia of the cross-section about the axis HB is 



D 

Fig. I. 



7 = 



(b-t)h- 



12 



(i) 



while the moment of inertia about CD has the value 

2fb' + ht' 



L = 



12 



(2) 



With these values of the moment of inertia, the general 
formula, M =-r, becomes (remembering that di=- or - 

bd^-(b-t)h^ 



or 



M=k 
M'=k 



6d 

2fb^-\-ht^ 
6b ' 



(3) 
(4) 



I 



Art. 106.] FLANGED BEAMS IVITH EQUAL FLANGES, 667 

k is written for all extreme fibre stress. 

Eq. (3) is the only formula of much real value. It will 
be found useful in making comparisons with the results 
of a simpler formula to be immediately developed. 

Let di=^{d+h). Since t' is small compared with 

-, the intensity of stress may be considered constant in 
2 

each flange without much error. In such a case the total 

stress in each flange will be kht' = Tbf, and each of those 

forces will act with the lever-arm ^di. Hence the moment 

of resistance of both flanges will be 

kht'-di. 

The moment of inertia of the web will be — . Conse- 

12 

quently its moment of resistance will have very nearly the 
value 

6 ■ 

The resisting moment of the whole beam will then be 

M=k(bt'dy^^^ (5) 

A further approximation is frequently made by writing 
d^h for h^\ then if each flange area ht' =/, eq. (5) takes the 
form 



M 



-kd^(f + f) (6) 



Eq. (6) shows that the resistance of the web is equivalent 
to that of one sixth the same amount concentrated in each 
flange. 



668 ROLLED ^ND CAST FLANGED BEAMS. [Ch. XIV 

If the web is very thin, so that its resistance may be 
neglected, 

M^kfdi^kbfdi, (7) 

or 

/=s <« 

Cases in which these formulas are admissible will be 
given hereafter. It virtually involves the assumption that 
the web is used wholly in resisting the shear, while the 
flanges resist the whole bending and nothing else. In 
other words, the web is assumed to take the place of the 
neutral surface in the solid beam, while the direct resistance 
to tension and compression of the longitudinal fibres of the 
latter is entirely supplied by the flanges. 

Again recapitulating the greatest moments in the more 
commonly occurring cases: 

Cantilever uniformly loaded, 

M = —=^ . (9) 

2 2 ^ 

Cantilever loaded at the end, 

M = Wl (10) 

Beam supported at each end and uniformly loaded, 

.M = -3-=^g-. ...... (II) 

Beam supported at each end and loaded at centre, 

Wl 

M = — (12) 

4 

Beam supported at each end and loaded both uniformly 
and at centre. 



Art. 107.] ROLLED 3TEHL FL.4NGED BEAMS. 669 

^=1(^ + 7) (^3) 

In all cases W is the total load or single load, while p, as 
usual, is the intensity of uniform load, and / the length of the 
beam. 

Art. 107. — Rolled Steel Flanged Beams. 

The resisting moments of all rolled steel beams sub- 
jected to bending are computed by the exact formula 

k being the greatest intensity of stress (i.e., in the extreme 
fibres) at the distance d^ from the neutral axis about which 
the moment of inertia / is taken. In all ordinary cases 
the webs of beams are vertical so that the axis for / is 
horizontal; but it sometimes is necessary to use the mo- 
ment of inertia / computed about the axis passing through 
the centre of gravity of section and parallel to the web. 
The latter is frequently employed in considering the lateral 
bending effect of the compression in the upper flange. 

The upper or compression flange of a rolled beam 
under transverse load, unless it is laterally supported, is 
somewhat in the condition of a long column and, hence, 
tends to bend or deflect in a lateral direction. This ten- 
dency depends to some extent on the ratio of the length of 
flange (/) to the radius of gyration (r) of the section about 
the axis parallel to the web, as will be shown in detail in 
a later article. It will be found there that the ultimate 
compression flange stress decreases as the ratio l^r in- 
creases. Hence in Table I there will be found values of 
l-^r for the different beams tested. 



670 ROLLED /iND C/fST FLANGED BEAMS. [Ch. XIV. 

The results of tests given in Table I were found by 
Mr. James Christie, Supt. of the Pencoyd Iron Co., and 
they are taken from a paper by him in the " Trans. Am. 
Soc. C. E." for 1884. All beams, both I and bulb, were 
loaded at the centre of span. Hence the moment of the 
centre load, W, and the uniform weight of the beam itself, 
pi, will be, as shown in eq. (13) of Art. 106, 

MJ-iw+i^)J4 (2) 

a\ 2/ di 



Hence 



4 
k 



=T/(^+?) ••••••• (3) 



The known data of each test will give all the quanti- 
ties in the second member of eq. (3). The two columns 
of elastic and ultimate values of k in the table were com- 
puted by eq. (3). The positions of the bulb beams (i.e., 
the bulb either up or down) in the tests are shown by the 
skeleton sections in the second column. 

The coefficients of elasticity E were computed from 
the data of the tests taken below the elastic limit by the 
aid of eq. (21), Art. 28: 

w+m, (4) 



4SE1 



W being the centre load and pi the weight of the beam, 
the length of span / being given in inches. 

All beams were rolled at the Pencoyd Iron Works. 
The ''mild steel" contained from o.ii to 0.15 per cent, of 
carbon, and the "high steel" about 0.36 per cent, of carbon. 
These steels are the same as those referred to in Art. 60. 

No. 14 is the only test of a "high" steel beam; all the 



Art. 107.] 



ROLLED STEEL FLANGED BEAMS. 



671 



remaining tests being with mild-steel shapes. Tests 3 to 
9 inclusive were of deck or bulb beams, as the skeleton 
sections show. 

Beams 3 and 4 were rolled from the same ingot, as were 
also 6 arid 7, as were also 10, 12, and 13, and as were also 
16, 17, 18, and 19. All beams were unsupported laterally 
in either flange. The moments of inertia were computed 
from the actual beam sections. The length of span is 
represented by /, while r is the radius of gyration of each 
beam section about an axis through its centre of gravity 
and parallel to its web. The values of r were as follows: 

5 inch I . . . .r=o.54inch. 3 inch I, 

6 " "....r = o.63 " 8 " " 

7 " "....r=o.7i " 10 " "....r = o.95 
9 " " r = o.83 " 12 " •• r = i.oi 

Table I. 
TRANSVERSE TESTS OF STEEL BEAMS. 



. .r=o.59 inch. 
. .r=o.88 " 













Final 


k in Pounds per 


Coefficient of 




Kind of 


Span 


/ 


Moment 


Centre 


Square 


inch at 


Elasticity E, 


No. 


Beam. 


in Ins. 




of 


Load in 
Pounds. 






in Pounds per 
Square Inch. 




r 


Inertia. 


















Elastic. 


Ultimate 


I M 


ild 3" I 


59 


100 


2.76 


5,500 


41,100 


45,200 


30,890,000 


2 


3" '' 


39 


66 


2.76 


8,300 


40,800 


45,100 


25,011,000 


3 


' s"? 


108 


200 


12 


8,800 


50,000 


55,000 


27,7 18,000 


4 


' 5"i 


108 


200 


12 


8,400 


46,900 


52,500 


25,489,000 


5 


' 6"2 


96 


152 


22 


14,860 


51,200 


54,300 


23,692,000 


6 


^"" 


69 


97 


37.6 


34,000 


47,100 


59,300 


18,765,000 


7 


' 7':? 


69 


97 


37.6 


34,000 


47,100 


59,300 


23,040,000 


8 


' 9"? 


240 


290 


84.8 


14,500 


46,000 


51,300 


29,923,000 


9 


' 9" I 


240 


290 


82.9 


13,500 


39,800 


48,800 


30,209,000 


10 


' 8" I 


240 


273 


70.2 


13,000 


37,600 


44,400 


28,889,000 


II 


8"" 


240 


273 


70.3 


12,930 


37,500 


44,100 


29,055,000 


12 


8" " 


144 


164 


70.2 


19,480 


32,800 


39,900 


31,313,000 


13 


' 8" " 


96 


109 


70.2 


31,300 


40,300 


42,800 


23,689,000 


14 H 


igh 3"" 


39 





2.74 


1 1,500 


54,300 


■ 


27,515,000 


IS M 


ild 10" " 


IS6 


164 


150.5 


22,500 


35,000 


• ■ 


28,414,000 


16 


10" " 


168 


177 


150.5 


21,000 


35,200 





27,182,000 


17 


! 10" II 


180 


189 


150.5 


19,500 


35,000 




29,160,000 


18 


10" " 


192 


202 


150.5 


18,000 


34,400 





29,727,000 


19 


' 12" II 


240 


238 


264.7 


24,500 


33,400 





30,749,000 


20 


1 2" " 


240 


238 


267.6 


24,200 


32,500 


■ 


29,568,000 


21 


' 12"" 


228 


226 


273-8 


22,000 


27-500 




29,164,000 


22 


' 12"" 


216 


214 


263.7 


29,000 


35,600 


• • 


30,219,000 


23 


' 12"" 


204 


202 


256.7 


27,000 


32,100 


. • 


30,030,000 


24 


' I2"|' 


192 


190 


257.8 


34,000 


38,000 


• 


29,709,000 


25 


' 12' " 


192 


190 


262.6 


34,000 


37,300 





28,234,000 


26 


' 12"" 


180 


178 


262 . 4 


36,700 


37,700 


• 


27,717,000 


27 


' I 2" " 


168 


166 


264.0 


38,000 


36,300 





28,784,000 


28 


' 12"" 


156 


154 


261.7 


43,000 


38,400 




27,818,000 



672 ROLLED AND Cy^ST FLANGED BEAMS. [Ch. XIV. 

The values of k both for the elastic limit and the ulti- 
mate are erratic, and the range of results in the table is not 
sufficient to establish any law, but on the whole the small 
ratios l^r accompany the larger values of k. The bulb 
or deck beams also appear to give larger values of k than 
the I beams. 

The results of these tests indicate that the greatest 
working intensities of stress in the flanges of rolled steel 
beams may be taken from 12,000 to 16,000 pounds per 
square inch if the length of unsupported compression 
flange does not exceed 15 or to 2 0or. 

In the work of design, the quantity 1 -^a'^ used in eq. (2), 
called the ''section modulus," is much employed, and it 
can be taken directly from the Cambria Steel Company's 
tables at the end of the book, as can the moment of inertia /. 
Eq. (2) shows that 

/ M . . 

j;--F- (5) 

Hence the moment of the loading in inch-pounds di- 
vided by the allowed greatest flange stress in pounds per 
square inch must be equal or approximately equal to the 
section modulus of the required beam. 

There may be found in the Proceedings of the Ameri(^an 
Society for Testing Materials, 1909, the results of tests 
of rolled I beams and girders produced by the Bethlehem 
Steel Company and of standard rolled I beams by Profes- 
sor Edgar Marburg. Also Professor H. F. Moore gives 
results of his testing of steel I beams of the regular or 
standard pattern in Bulletin No. 68 of the University of 
Illinois. Professor Marburg's main purpose appears to 
have been to make comparative tests of the ordinary 
I beam and of the wide-flange Bethlehem shapes, while 
the principal object of Professor Moore was to investigate 



Art. 107.] 



ROLLED STEEL ELAN GEO BEAMS. 



673 



the influence of lateral deflection on the capacity of the 
compressive flange without lateral support. Table II 
gives the results of these tests, each of professor Marburg's 
results except one being an average of three. 

Table II. 

TESTS OF ROLLED STEEL BEAMS. 





Span, I 
Ft. 


I 
r' 


Extreme Fibre Stress 
k, 'Lbs. per Sq.in. 


Modulus of 
Elasticity. 


Size. 


Type. 


Bias. 
Limit. 


Ultimate. 


Beth. I 

Std. I 

Girder 

Beth. I 

Std. I 

Girder 

Beth. I 

Std. I 

Girder 

Beth. I*. . .. 
Girder ...... 


15 
15 
15 
15 
15 

11 

20 

' 20 

20 

20 


125 
167 

75 
125 
167 

75 
129 
176 

90 
III 

84 


31,700 
' 20,400 
26,700 
21,800 
20,600 
22,500 
20,900 
19,500 
15,400 
13,000 
11,800 


46,100 
42,200 
53,900 
37,900 
34,700 
41,100 
34,600 
33,000 
34,300 
32,300 
31,000 


26,900,000 
26,200,000 
26,900,000 
26,406,000 
26,900,000 
27,200,000 
26,400,000 
25,800,000 
25,600,000 
29,400,000 
24,800,000 


15" 38 1b. 
15" 42 " 
15" 73" 
15" 38" 
15" 42 " 

15" 73 " 
24" 72 " 
24" 80 " 
24" 120 " 
30" 120 " 
30" 175 " 



* One beam only. 

Prof. Moore's fifteen tests were with 8-inch, 18-pound and one 25-pound I beams, 
the spans being 5. 7-5. 7-92, 10, 15. i5 7 and 20 feet. The ratio l-^r' varied from 71 to 
286. The ultimate fibre stress k was Max. 36,600; Mean 32,300; Min. 28,100. The 
Modulus E was. Max. 32,300,000; Mean 28,400,000; Min. 25,100,000. The Max. E is 
to be regarded with doubt. 

As is the case with all tests of full-size rolled beams, 
the results are seen to vary quite widely. This is largely due 
to the fact that such full -size members are seldom true 
in all their parts, i.e., the web may be a little twisted on the 
cooling bed and the flange will perhaps n3ver be perfectly 
plane, consequently the applied load in the testing machine 
will not be received with presupposed exactness. Again, 
the work of the rolls and the effects of cooling will not be 
uniform. At any rate the most scrupulous care in testing 
will not prevent many erratic results, apparently unac- 
countable. 

In order to show these results graphically they have 



674 



ROLLED AND CAST FLANGED BEAMS. [Ch. XIV. 



been .plotted on Plate I and the explanatory matter on 
the Plate will make clear the results belonging to each 

investigator. The horizontal ordinate is the ratio — , r' 

being the radius of gyration of the normal section of 
the column about a vertical axis parallel to the web and 
passing through its centre. The vertical ordinate is the 
intensity k of the extreme fibre stress produced by the 
ultimate load on the beam as shown in Table I. 
The equation 

^=39,000-44-, 



represents the broken line drawn on Plate I. It is a tenta- 



-50-000 








- 












' 


^ - 






PI 


atel 






H 








• 













































• 




40-000 






•1 




• 






X 










: 












__ 


T 




TtT 


T 


















30-000 








•1 
•1 









.X 




-- .. 


x 




























-5- 








T^ 




20-000 






























































10-000 








Bethlehem I ? „ .. 
V Girder \ ^^^^"^^ 
X Standard I 
— Moore 




































• 


Chri 


stie 
000 - 


"i 



















/-VJ-'=40 



120 



200 



220 



tive expression, as there are not sufficient tests with the 
requisite variation of — to justify more than a trial value of k. 



Art. 107.] ROLLED STEEL FLANGED BE/IMS. 675 

Professor Marburg made no effort to give lateral sup- 
port to his beams under test, nor did he endeavor to give 
the compressive flange lateral freedom, as did Professor 
Moore for a part of his tests. As, however, the results 
appear to be about the same, whether the compressive flange 
has complete lateral freedom or not, under ordinary cir- 
cumstances of testing, no distinction is made on this account 
between the various plottings on Plate I. The extremely 
high values on that Plate belong to the first nine tests 
by Mr. Christie, as given in Table I. They are abnormally 
high and whether such results are characteristic of bulb 
sections or due to some other reason is not clear. 

Prob. I. It is required to design a rolled steel beam 
for an eftective span of 20 ft. to carry a uniform load of 
725 lbs. per linear foot in addition to the weight of the beam 
itself, the circumstances being such that it is not advis- 
able to use a greater total depth of beam than 12 ins. 
The greatest permitted extreme fibre stress k will be 
taken at 12,000 lbs. per sq. in. It will be assumed for 
trial purposes that the beam itself will weigh 35 lbs. per 
linear foot, so that the total uniform load will be 760 lbs. 
per linear foot. The centre moment in inch-pounds will, 
therefore, be 

^^ 760X20X20X12 ^ . „ 

J\I =■ ^ =456,000 m. -lbs. 

By eq. (5) the section modulus will be 456,000^-12,000 
= 38. By referring to the tables in almost any steel com- 
pany's handbook it will be found that this section modulus 
belongs to a 12-inch, 35-pound steel rolled beam, and 
that beam fulfills the requirements of the problem. 

Prob. 2. It is required to design a rolled-steel beam 
for a 3 2 -ft. effective span to carry a load of 1280 pounds per 
linear foot in addition to the weight of the beam, and a 



676 ROLLED /IND CAST FLANGF.D BEAMS. [Ch. XIV. 

concentrated load of 1 1 ,000 pounds at a point 1 1 feet distant 
from one end of the span. The greatest permitted work- 
ing stress in the extreme fibres of the beam is 16,000 lbs. 
per sq. in. 

It will be assumed for trial purposes that a 24-in. beam 
weighing 95 lbs. per linear foot will be required so that 
the total uniform load per linear foot will be 1375 pounds. 
It will then be necessary to ascertain at what point in the 
span the maximum bending moment occurs, i.e., at what 
point the transverse shear is equal to zero. Let a be the 
distance of the concentrated weight from the nearest end 
of the span, i.e., a = 11 ft. Then 'et P be the single weight, 
p the total uniform load per linear foot, and / the length 
of span. The following equation representing the condi- 
tion that the transverse shear must be equal to zero may- 
be written 

pi Pa 

Hence x = - +—7 . 

2 pi 

In the above equation x is obviously the distance from 
that end of the span farthest from P to the section of 
greatest bending moment. Substituting the above numeri- 
cal values in the equation for x, there will result 

:r = i6 + 2.75-i8.75 ft. 

Since 32 — 18.75=13.25 the following will be the value 
of the greatest bending moment in inch-pounds: 



.^ y 1375X18.75 11,000X11 „ , 
M^y-^^ ^X 13.25 + X 18.75 j 12 

= 2,900,363 inch-pounds. 



Art. io8.] DEFLECTION OF ROLLED STEEL[ BEAMS, 677 

The section modulus of the beam required is by eq. (5) 
2,900,363^16,000 = 181. The section modulus of a 24. -in. 
steel beam weighing 85 lbs. per linear foot is 180.7, a-s will 
be found by referring to the tables at the end of the book. 
Hence that beam will be assumed for the correct solution 
of the problem. The fact that the beam w^eighs 10 lbs. 
per linear foot less than the assumed weight has too small 
an effect upon the greatest bending moment to call for 
any revision. 

Prob. 3. A steel tee beam of 8 ft. span is to be used as 
a purlin to carry a uniform load of 125 lbs. per linear foot 
with the web of the tee in a vertical position. The greatest 
permitted intensity of stress in the extreme fibre of the 
tee is 14,000 lbs. per sq. in. It is required to find the 
dimensions of the tee. By referring to eq. (5) the section 
modulus will be written 

1000X96 Q.. 

S = ^- = . 86 m. 

0X14,000 

By referring again to the steel handbook tables it 
will be found that a 3 X3 XA in. steel tee weighing 6.6 
lbs. per lin. ft. has just the section modulus required. 
That tee therefore fulfils the requirements of the problem. 

Prob. 4. It is required to support a single weight of 
12,000 lbs. at the centre of a span of 13 ft. 6 ins. on two 
rolled steel channels with their webs in a vertical position 
and separated back to back by a distance of 3 ins., the 
greatest permitted intensity of stress in the extreme fibre 
of the flanges being 15,000 lbs. Find the size of channels 
required. 

Art. 108.— The Deflection of Rolled Steel Beams. 

The deflections of rolled steel beams may readily be 
computed by the formula of Art. 28. The general pro- 



678 ROLLED AND CAST FLANGED BEAMS. [Ch. XIV. 

cedure will be illustrated by using the equations for a non- 
continuous beam simply supported at each end and loaded 
by a weight at the centre of span, or uniformly, or in both 
ways concurrently. Eq. (20) will give the deflection at 
any point located by the coordinate x, while eq. (21) will 
give the centre deflection only. The tangent of the in- 
clination of the neutral surface at any point located by x 

dzv 
will be given by the value of -t~ found in eq. (19). 

Prob. I. Let the centre deflection of the rolled-steel 
beam of Prob. i of Art. 107 be required. Referring to 
eq. (21) of Art. 22, 

W = o; /== 20 feet = 240 inches; /? = 760 pounds; 
7 = 228.3; and E may be taken at 29,000,000. 

Hence the centre deflection is 

240X240X240X5X760X20 . . 

^^ = 48X8X29,00-0,000X228.3 = • 4'4 mch. 

If half the external uniform load of 725 pounds per 
linear foot had been concentrated at the centre of span, 



^^_ 725X20 , 

W =-^—^ ■ =7250 pounds; p = 



35 



2 

and I = 20 ft. =240 ins. Also pi = 700 pounds. 

Hence the centre deflection would be 

240 X 240 X 240 X (72 50 + 437 • 5) ..,• ^-u 

w, = :^ 7; = . ^ ^ ^ men. 

1 48X29,000,000X228.3 ^^^ 

Prob. 2. In Prob. 2 of Art. 107 place the 11,000-pound 
weight at the centre of span, then find the inclination of 
the neutral surface and the deflection of the 24-inch 85- 



Art. 109.] IVROUGHT-IRON ROLLED BEAMS. 679 

pound steel beam at the centre and quarter points of the 
32-foot span, taking £^ = 29,000,000 pounds. 

Art. 109. — Wrought-iron Rolled Beams. 

Although wrought-iron rolled beams are not now manu- 
factured, being cofnpletely displaced by steel beams, yet 
many are still in use. Hence it is advisable to exhibit 
the empirical quantities required to design them and to 
determine their safe carrying capacities as well as their 
deflections under loading. 

It has been observed in Art. 107 that the upper or com- 
pression flange of a loaded flanged beam will deflect or 
tend to deflect laterally at a lower intensity of compressive 
stress as the unsupported length of such a flange is in- 
creased. The experimental results given in Table I ex- 
hibit the values of the intensity of stress K in the extreme 
fibres of the beam both at the elastic and ultimate limits, 
the usual formula for bending resistance being used, 

«=f <-) 

In the autumn of 1883 an extensive series of tests of 
wrought-iron rolled beams, subjected to bending by centre 
loads, was made by the author, assisted by G. H. Elmore, 
C.E., at the mechanical laboratory of the Rensselaer Poly- 
technic Institute. The object of these tests was to dis- 
cover, if possible, the law connecting the value of K for 
this class of beams with the length of span when the beam 
is entirely without lateral support. The means by which the 
latter end was accomplished, and a full detailed account 
of the tests will be found in Vol. I, No. i, " Selected Papers 
of the Rensselaer Society of Engineers." The main results 
of the tests are given in Table I. All the tests were made 
on 6 -inch I beams with the same area of normal cross- 



68o 



ROLLED AND CAST FL/INGFD BEAMS. 

Table I. 



[Ch. XIV. 











K 












Final 








Perm'nent 


Perm'nent 


E 


No 


Span, 


Centre 


/. 






Vertieal 


Lateral 


Pounds per 
Square Inch. 




Feet. 


Weight, 
Pounds. 


r 


Elastic 
Limit, 


Ultimate, 
Pounds. 


Deflection, 
Inches. 


Deflection, 
Inches. 










Pounds. 










I 


- 20 


4,060 


400 


27,726 


31,094 


0.14 




24,170,000 


2 




4, 200 


400 


29,623 


32,885 


0.30 




26,374,000 


3 


I ^^ 


4.390 


360 


28,264 


30,791 


0.2 


0.5 


24,520,000 


4 


4,570 


360 


28.264 


32,020 


0.18 


0.4 


24,313,000 


5 


[i6 


4,770 


320 


26,564 


29,579 


0.28 


1. 00 


25,771,000 


6 


5,270 


320 


29,596 


32,632 


0.48 


1.25 


25,003,000 


7 


[14 


6,130 


280 


31,191 


33,049 


0.30 


I .20 


26,082,000 


8 


6,125 


280 


31,164 


33,023 


0.30 


I .10 


23,373,000 


9 


/■ 1 2 


7,161 


240 


30,221 


32,907 


0.35 


1.08 


25,287,000 


lO 




7,350 


240 


31,314 


33,817 


0.33 


I .09 


24,022,000 


II 


i ^° 


9,255 


200 


33,082 


35,358 


0.39 


1.08 


25,115,000 


12 


9,655 


200 


33,082 


37,064 


0.50 


1.50 


24,218,000 


13 


[« 


11,485 


160 


29,736 


35,010 


0.30 


0.90 


21,61 1,000 


14 


11,980 


160 


31,936 


36,527 


0.29 


I .05 


21,987,000 


15 


[ ^ 


18,300 


120 


35,497 


41,737 


0.605 


1-53 


23,040,000 


i6 


18,145 


120 


36,617 


41,396 


0.67 


1.88 


20,935,000 


17 


• 5 


22,870 


100 


34,136 


43,434 


0.67 


1-75 


22,023,000 


i8 


23,065 


100 


34,136 


43,813 


0.67 


1-75 


25,272,000 


19 


!- 


29,985 


80 


32,619 


45,532 


0.96 


1.70 


24,315,000 


20 


28,585 


80 


32,619 


44,744 


0.60 


1.86 


21,275,000 



section of 4.35 square inches. Actual measurement showed 
the depth d of the beams to be 6.16 inches. The moment 
of inertia of the beam section about a line through its 
centre and normal to the web was 7 = 24.336. The radius 
of gyration of the same section in reference to a line through 
its centre and parallel to the web was r = o.6 inch. / was 
the length of span in inches. 

If M is the bending moment in inch-pounds, W the 
total centre load (including weight of beam), and K the 
stress per square inch in extreme fibre, the following 
formulas result: 



k^ 



Md 
2I 



and M = 



Wl 



k = 



Wld 
8/ • 



(2) 
(3) 



Art. 109.] 



IVROUGHT-IRON ROLLED REAMS. 



681 



The experimental values of II \ /, d, and / inserted in 
the above formula give the values of k shown in the table. 
The coefficient of elasticity, E, was found by the usual 
formula, 

in which w is the deflection caused by W. 

The full line is the graphical representation of the values 
of k given in Table I. Since k must clearly decrease with 





PidLe I. 


1 1 1^ 'i ' . 1 1 ■" ' ! — ■■ ~ ~1 ' ■; rn 1 'T- ■■ ■ III 1 


M i M 1 1 1 1 i ' ' i : 


1 i t - 1 11 r- -r ]-■■■■ 1 t 1 


snnt a 1 V- 1 M ' ! ■ ' ' 












! ; 1 !>^>^^^' ' ■ ' , , , Ml, I ■ i M i 1 1 M M M ■ I M I 1 11 1 i -y- --1 








,,,nfin ■ ' Ml' ' 1 M ' fSt'-i.' ' i ' i 1 II ' ' ' ' ' ' ' 1 1 M ' ! 1 ' M ' ' ' Mi' ' 


^OOi 1 Mil i , M 1 1 r'^-i>^. Mm ' ' M 1 i M i M 1 1 1 MM ■ M M 1 i 


1 M M ' ' ' 1 ' '^~^riS-.' 11 M M 1 1 M ' M 1 1 1 1 1 1 1 1 


1 \ \ \\ Ml' ' ' M M M^-^-'t. M 'Ti M ' ' M M 1 


i MM Ml. . , 1 1 i ! >>T>>J V 1 1 1 M 1 ■ 1 


■ ■ --'■ 1 i 1 , : \U\ i^i'SjS-. 1 Mil'' 1 1 1 


. 1 • ' M •■ ' 1 i ' ' ' ' ■ ' rKjJs^i 1 M 1 1 ' : : M 1 1 1 M 




-r^^ nVH --^^^U^ ' -M4+ + ^^ H-- 


\ 'rn h 


i 1 1 1 III' (flT^ r^-M-^ ' Wl^jAl ' M ■ ' 1 'ii 


cUUUU- ■ III 1 


LllJl._--_-_--------l-±-^4---.-^.^^4± im^ irl -:^#F^ 




- M M M M 1 M 1 LI M 1 " ■'■ ' 1 1 M i ' 1 i M 1 M 1 1 ■ + 


t 1 1-L M \\\ M iff> a^n M'tiO 1 \ '"■n "on i M ' ^^p M \m 





the length of span, and increase with the radius of gyration 
of the section about an axis through its centre and parallel 
to the web (the latter, of course, being vertical), k has 
been plotted in reference to l^r as shown. No simple 
formula \yill closely represent this curve, but the bioken 
line covers all lengths of span used in ordinary engineering 
practice, and is represented by the formula 



^=51,000-75- 



(5) 



For raiVay structures the greatest allowable stress 
per square inch in the extreme fibres of rolled beams may 
be taken at 

^ = 10,000-15--. (6) 



682 ROLLHD AND CAST FLANGED BEAMS. [Ch. XIV. 

Values of k taken from a large scale plate, like Plate I, 
are, however, far preferable to those given by any formula. 

The ultimate values of k given in Table I are fairly 
representative of the best wrought-iron I beams. The 
coefficients of elasticity E range from about 22,000,000 to 
about 25,000,000 pounds; the average may be taken about 
24,000,000 pounds. 

The deflection of wrought-iron beams may be computed 
by the formula 

WP . . 

^ = ^8E7' ^^) 

when the load W is at the centre of the beam. In the 
general case of a beam carrying the centre load W and 
the uniform oad pi, the quantity {W -^Ipl) must displace 
W in eq. (7). If the beam carry only the tiniform load pi, 
W in eq. (7) must be displaced by \pl. 

If it is desired to apply the law expressed in eqs. (5) 
and (6) to mild-steel beams, the second members of those 
equations may be multipHed by | to f for close approxi- 
mations. 



CHAPTER XV. 

PLATE GIRDERS. 

Art. no. — The Design of a Plate Girder. 

A PLATE girder is a flanged girder or beam built usually 
of plates and angles, the flanges being secured to the web 
by the proper number of rivets suitably distributed. The 
flanges, unlike those of rolled beams, are usually of vary- 
ing sectional area, although occasionally either flange may 
be of uniform section throughout when formed of two 
angles, or two angles and a cover-plate. Fig. i is a general 
view of a plate girder, while Figs. 2, 3, 4, and 5 show 
some of the general features of design. 

The total length of a plate girder is materially more 
than the length of clear span over which the girder is de- 
signed to carry load. Blocks or pedestals of masonry or 
metal, as the case may be, support the ends of the girders 
and rest on the masonry or other supporting masses or 
members carrying the girder and its load. The distance 
between the centres of these blocks or pedestals is called 
the effective span of the girder, as it is the span length 
which must be used in computing bending moments, 
shears, or reactions. Plate girders must evidently be 
somewhat longer than the effective span. >In the Figs, the 
relations of the various parts at the end of the plate girder 
are shown in detail. The girder illustrated in Fig. i has 

683 



684 PLATE GIRDERS. [Ch. XV. 

an effective span of 68 ft. with the centre of the pedestal 
block 15 inches from the face of the masonry abutment 
and 12 inches from the extreme end of the girder. The 
effective depth of the girder is the vertical distance or 
depth between the centres of gravity of the two flanges. 
When the girder has cover-plates this effective depth may 
be greater than the depth of web plate at the centre of 
span and less than that at the ends, even when the web 
plate is of uniform depth. It is always customary, how- 
ever, to take the effective depth of a plate girder with 
uniform depth of w^eb as constant. Frequently that depth 
is taken equal to the depth of the web plate; or, again, it 
may be taken equal to the depth between the centres of 
gravity of the flanges at mid-span without sensible error. 
In case the web plate is not of uniform depth the effective 
depth might still be taken as the depth of web plate at 
the various sections of the girder, or it may be taken as the 
depth between centres of gravity of the flanges at the same 
sections. 

The plate girder shown in Fig. i and to be assumed for 
the purposes of design is of the deck type and has a clear 
span of 65 ft. 6 ins., an effective span of 68 ft., and a length 
over all of 70 ft. The differences between the effective 
span and the clear span and total length are obviously 
dependent upon the length of span. For short spans 
those differences are relatively small, and relatively large 
for long spans. The depth of w^eb plate will be taken as 
6 ft. 8 ins., and it will be found later that at and in the vicin- 
ity of the centre of span three cover-plates will be needed. 
The girder will be assumed to be of mild structural steel 
and will be supposed to carry a single-track railroad mov- 
ing load with the concentrations and spacings shown in 
Table I, Art. 21. 

The dead load or own weight of the girder and track 
will depend somewhat upon whether the girder is of the 



^1-7J<^-* 



i-li--ipr-i:i)- 







, Sole PI. ll"x ^ X 1 6 PC 



•Remainder of Bott. li 
as Top Flange excexi 
' holes to be shop rivi 



,3* . 5K 



»l«l*C*|<-*k 



-IVA 



4% 



4_|_l_c4£!f.^, 



pQ-ffl-l 

'6-4-1 



i 



-(p-OffiO oo- 



3' „5>, 



O— 0<P (pSO— (!>- 



3-llJ^- 



\-T-T-t-T-9' 



H-|-oo-i"o-o6-H-l- 
l-f-|-6i-;i-f-f<&-H-H 



-3-11}^ >UrO^-»|« ZV/^* 

0-666 o o-fio-ooooo-o o 

• " |:^:t-i 




{To face page 6'oS-^ 



Art. no.] THE DESIGN OF A PLATE GIRDER. 685 

through or deck type. The only difference in computa- 
tion arising in those two types is due to the fact that if the 
girders are of the deck class (i.e., carrying the moving load 
directly on their upper flanges) the rivets connecting the 
upper flanges with the webs must be assumed to carry the 
wheel concentrations in addition to their other duties, as 
will be shown in the following computations. The total 
dead load or own weight will be taken as 1400 lbs. per linear 
foot. Inasmuch as there are two girders, each will carry 
one half of the moving load and one half of the dead load 
or own w^eight. It should be observed that the effective 
length of span being 68 ft., the two locomotives at the head 
of the train load will more than cover the span, so that the 
uniform train load will not appear in the computations. 

The design of this plate girder will be made in accord- 
ance with the provisions of the American Railway Engi- 
neering and Maintenance of Way Association and refer- 
ences will be made to those provisions. 

Bending Moments. 

The first computations necessary are those required 
to determine the bending moments, and from them the 
flange stresses at different points of the span. Those 
points may be taken at 5, 8, or 10 ft. apart as may be 
desired for the purpose of design; the closer together the 
sections are taken the greater will be the degree of accuracy 
attained. In the present instance those sections will be 
taken 5 feet apart up to 25 ft. from the end of the span, 
but the next or final section will be at the centre of span. 
After the bending moments are obtained, the flange 
stresses at once result by dividing the former by the efl^ec- 
tive depth. 

Figs. I and 2 show the complete single-track railway 



686 PLATE GIRDERS. [Ch. XV. 

deck-plate girder span consisting of two girders with the 
requisite bracing connections between them. The total 
dead load or own weight is a uniform load and consists of: 

Lbs. per L-n. Ft. 

Track (ties, rails, etc.) 450 

Two girders and bracing 1050 

Total 1500 

Or for one girder -^ — = 750 

2 

As each girder will carry 750 lbs. of dead load per linear 
foot, and as the effective span is 68 ft., the expression for 
the dead-load bending moment in foot-pounds at any 
point will be as follows : 

M=^^(68x-x2) (i) 

2 

The application of eq. (i) to the sections of the girder 
5, ic, 15, 20, 25, and 34 ft. from the ends will give the 
following expressions for the bending moments in foot- 
pounds : 

D. L. Moment. 
X Ft. Lbs. 

5 . . . . .' 118,120 

10 • . 217,500 

15 298,100 

20 360,000 

25 403,100 

34 433,500 

The moving-load bending moments are next to be found 
by using the concentrations shown in Table i, Art. 21. 
For this purpose the criterion for the maximum bending 



Art. no.] THE DESIGN OF A PLATE GIRDER. 687 

moment, cq. (5), Art. 21, must be applied at the assumed 
sections in which V (equal to x in the above dead-load 
computations) has the values 5, 10, 15, 20, 25, and 34 ft. 
The application of that criterion to the section BO, Fig. i, 
5 ft. from the end of the span shows that W2, or the first 
driving wheel, must rest at the section in question for the 
maximum bending moment, the loads W2 to 1^12 inclusive 
resting on the span. Wi will be off the span. By the aid 
of Table i, Art. 21, the greatest bending moment desired 
is: 

Ms =^^(9,030,000 + 2 X 273,000) =704,000 ft. -lbs. 
68 

Similarly for the section CN, 10 ft. from the end of the 
span, the criterion eq. (5) of Art. 21 shows that 1/^3 must 
be placed at C with W12 2 ft. from the end of the span 
and Wi off the span. By the aid of Table i the desired 
moment takes the value: 

Mio= — (9,030,000-^-2 X273,ooo) — 150,000= 1,260,000 ft. -lbs. 
60 

Concisely stating the conditions and results for the 
remaining sections shown on Fig. i: For DL, 15 feet 
from end of span, two positions of moving load, I/F3 at D 
and W12 at D satisfy the criterion, but the latter with 
13 feet of uniform train load on the span gives the greatest 
moment. Total load on the span is 

(1^10+ . . . +VF18+3000X13) 
and the moment is: 

^15=— (6, 3 10, 000 +2 13, 000X13 +3000 X — j -345,000 = 

1,715,000 ft. -lbs. 



088 PLATE GIRDERS. [Ch. XV. 

For EM, 20 feet from end of span, place 1^12 at E\ 

M20 ==^(6, 310, 000+8(213, 000 H 3£oo\\ _^^^ QQQ^ 

2,040,000 ft. -lbs. 

For GH, 25 feet from end of span, place W12 at G and 
the moment is : 



M 



25 



25/ 



000+232,500X3+3000—) -755.000 = 

2,265,000 ft. -lbs. 



The moment at the centre of the span can be computed 
in the same manner, but by referring to Table II of Art. 
2 1 , it will be seen to be : 

M34 = 2,435,400 ft. -lbs. 

A reference to the American Railway Engineering 
and Maintenance of Way Association specifications, Art. 
9, will show that the required allowance for impact is 
represented by the factor 7, in which V is the length of 
load on the span: 



1=^ 



300 



L'+30o* 



The positions of loading already found for the greatest 
moving load moments give the lengths V in feet in the 
following table: 



Pt. 


Loaded 
Length, L '. 


Impact 


Moving Load 


Impact Moment 


Ft. 


Factor I. 


Moment, Ft. -lbs. 


Ft.-lbs. ■ 


5 


63 


.827 


704,000 


582,000 


10 


63 


.827 


1,260,000 


1,040,000 


15 


66 


.820 


1,715,000 


1,405,000 


20 


61 


.897 


2,040,000 


1,830,000 


25 


64 


.825 


2,265,000 


1,866,000 


34 


68 


.815 

• 


2,435,000 


1,985,000 



Art. no.] THE DESIGN OE A PLATE GIRDER. 689 

By adding the dead load or own weight moments, 
already computed, to the moving load and impact moments 
in the preceding table, the total or resultant moments 
will be: 

Table I. 

T34- Total Moment 

^^- Ft.-lbs. 

5 1,404,000 

10 2,518,000 

15 3,418,000 

20 4,230,000 

25 4,534,000 

34 4,855,000 

Shears. 

Both dead and moving load shears must be computed. 
As the dead load or own weight is a uniform load on the 
girder, the shear at any point is simply the load between 
that point and the centre of span. Hence indicating the 
transverse shear at any section by the figure showing its 
distance from the end of the span, there will result the 
following values, 5o being the end shear or reaction: 

5o =34 X750 =25,000 lbs. 

S5 = 29 X750 = 21,750 
5io = 24 X750 = 18,000 
5"i5 = i9X75o = i4,25o 

520 = 14X750 = 10.500 
525= 9X750= 6,750 
534= 0X750= o 

The moving load shears will also be needed. Although 
there is no systematic criterion for such shears at different 



690 PLATE GIRDERS. [Ch. XV. 

points in a span traversed by a train of concentrations, 
it is a simple matter to find the greatest moving load shears 
at the sections contemplated by inspection and trial. The 
greatest end shear, i.e., the greatest reaction, has been 
found in Art. 21 and is given in Table II of that Article: 

End shear for 68-ft. span = 161,700 lbs. 
End impact shear =131,800 '' 

The impact factors for the shears are computed by the 
same formula already used for impact moments. 

For a shear 5 feet from end of span: place W2 at the 
5 -foot section, then the greatest shear is 

c, 9,030,000 + 2X273,000 ., 

55 = ^ ^ 7^ -^ = 141,000 lbs. 

Do 

By trying other positions it will be found that this 
gives the greatest shear. Wi is not on the girder and W12 
is 2 feet from the end of the span. 

For section 10 feet from end: place Wn at the section. 
Hence 

6,310,000 + 213,000X13 -1 3000X-— 

5 0=- ^^ ^ -150.000 = 

122,000 lbs. 

For section 15 feet from end: place Wn at the section 
r.nd there will result 

g2 

6,310,000 + 213,000X8 +3000 X — 
S , = = 1 so, 000 = 

104,300 lbs. 



Art. no. 



THE DESIGN OF A PLATE GIRDER. 



691 



For 20-ft. section: place W2 at the section and there 
will result 

c^ 6,QS0,000 ,, 

020 = ^^-^ 150,000 =87,200 lbs. 

68 



For a 2 5 -ft. section: place W2 at the section and the 
greatest shear will be 

^ S. 240, 000 + 213, 000 X3 11 

525=^^-^*^^ T^ ^-150,000 = 71,500 lbs. 

Oo 

For the centre of span : place 1^2 at that point and the 
greatest shear will be : 

^ ^,230,000 + 174,000X5 1, 

534=^^^-^^^ t:^ ^-150,000=45,300 lbs. 

Oo 

The loaded lengths in each of these cases to be used 
in computing the impact factors are in the order of the 
sections beginning with that at 5 feet from the end, 63, 
66,. 61, 56, 51, and 42 feet, the latter belonging to the 
centre of span. The following tabular statement repre- 
sents the elements of these moving load shears and the 
impact allowances: 

SHEARS AND IMPACT ALLOWANCES 



Section. 


Loaded 


Impact 


Moving Load 


Impact Shear. 


Length. Ft. 


Factor. 


Shear. Lbs. 


Lbs. 


5 


63 


.8.7 


141,000 


116,500 


10 


66 


.820 


122,000 


100,000 


15 


61 


■831 


104,300 


86,600 


20 


56 


.824 


87,200 


71,800 


25 


51 


•855 


71,500 


61,100 


34 


42 


•877 


45,300 


39,700 



692 PLATE GIRDERS. [Ch. XV. 

Adding together the dead load, moving load and impact 
shears as now determined, the following will be the resultant 
or total shears at sections under consideration: 

Table II. 

RESULTANT OR TOTAL SHEARS. 

Section. lotal^hears. 

End . 319,000 

5 279,300 

10 240,500 

15 205,100 

20 169,500 

25 139,400 

34 85.000 

The preceding results or computations due to the dead 
and moving loads are the principal data required in the 
design of the girder. 

Weh Plate. 

The effective depth of the girder will tentatively be 
taken as 6 feet 8 inches and the depth from the back of 
flange angles in the upper flange to the back of the lower 
flange angles will be taken as 6 feet 8| inches. As the 
depth of the web plate must be taken a little less ihan the 
depth from back to back of angles, in order that the flange 
plates may not touch the edges of the web plates when 
the different parts of the girder are assembled, that depth 
should be taken as 6 feet 8 inches. In fact the effective 
depth of a plate girder is sometimes prescribed as the depth 
of the wxb plate. This depth of web plate will leave 
\ inch clear at the top and bottom flanges, which is sufficient 
to insure the flange plates freedom from hitting the edges 
of the web. 

Art. 18 of the Specifications allows a working stress 
in shear of 10,000 pounds per square inch of gross cross- 



Art. no.] THE DESIGN OF A PLATE GIRDER. 693 

section of the web. As the total end shear has been 
found to be 319,000 pounds, the gross web plate section 
at the end of span should be 31.9 square inches. The 

minimum thickness must then be '-^^7^ = .399 inch. 

80 

A web plate 80 X-^ inch will be used, giving a gross 
16 

sectional area of 80 X. 43 75 =35 square inches. The sur- 
plus area is small and it is judicious design to have it. 
This web plate thickness also satisfies Art. 29 of the Speci- 
fications which prescribes that ' ' The thickness of web 

plates shall not be less than of the unsupported dis- 

160 

tance between fiange angles," as 6X6 inch flange angles 

will be used, 



Flanges. 

Art. 29 of the Specifications provides that the design 
of the flanges may be based either on the moment of inertia 
of the net section of the girder or on the assumption that 
the flange stress is of constant intensity with its centre 
at the centre of gravity of the flange area, the latter 
including one-eighth of the gross section of the web, the 
difference between one-sixth and one-eighth of the w^eb 
section being supposed to cover the material punched out 
in the tension side of the web plate. The latter method 
will be employed. 

Art. 30 of the Speciflcations provides that '' The gross 
section of the compression flanges of plate girders shall 
not be less than the gross section of the tension flanges." 
It will be best, therefore, to design the tension flange 
first. 

Using the total or resultant bending moment at the 



694 PLATE GIRDERS. [Ch. XV. 

centre of the span, the trial effective depth of 6 feet 8 inches 
will give the centre flange stress as follows: 



4,855,000 „ ,. 

- ^^' = 728,000 lbs. 

6.07 



The specifications permit a working tensile stress in 
the net section of the tension flange of 16,000 pounds per 
square inch. Hence the required net tension flange area 
is 

728,000 



16,000 



= 45.5 sq.ms. 



The available flange section due to one-eighth the gross 

•7 r 

sectional area of the web is ^ =4.375 square inches. The 

8 

amount of flange area to be supplied by the flange plates 

and angles is, therefore, 

45-5 -4-4 =41-1 sq.ins. . 

In providing 41. i square inches it is necessary to know 
what rivet holes are to be deducted from each cover-plate 
and each flange angle. It is clear that two rivet holes 
only need be deducted from each cover-plate, and it is plain 
that at least two rivet holes must be deducted from each 
flange angle section. In designing cover-plates for flanges 
it must be remembered that no such plate must be thicker 
than the one under it, i.e., if these plates are not of the same 
thickness, the thickest one must lie on the angles, the 
remaining thicknesses -to decrease or be the same in passing 
outward from the angles. As a trial section let the follow- 
ing be assumed: 



I 



Art. no. 



THE DESIGN OF A PLATE GIRDER 



695 



Angles or Cover-plates. 


Gross Area. 
Sq.Ins. 


Less Rivet Holes. 
Sq.Ins. 


Net Section. 
Sq.Ins. 


2 6"X6"xr' 

3 covers 14" Xl". .. 


16.88 
315 


4XiX!=30 
6XiXf=4-5 


13-88 
27.00 






48.38 • 


40.88 





As 40.88 square inches is but ij per cent, less than the 
desired area, 41.4 square inches, the former may be accepted 
subject to further confirmation. 

If the centre of gravity of the gross section of the tenta- 
tive flange area consisting of the three plates and two 
angles indicated above be determined, it will be found 
.11 inch above the back of the angles. This will make the 
effective depth 

6 ft. 8.5 ins. +.22 in. = 6 ft. 8.72 ins. 

This increase in effective depth will correspondingly 
decrease the centre flange stress so as to make the total 
actual net area of 45.3 square inches a little larger than 
required. Hence the trial centre tension flange area as 
determined above will be accepted as the actual flange area 
to be used, i.e., three i4Xi-inch cover-plates and two 
angles 6 X 6 X | inch. 



Length of Cover-plates. 

In the next Article there will be shown two methods 
of determining the lengths of cover-plates after the 
sections of those plates have been found for the greatest 
bending moment, usually taken as at the centre of span. 
These two methods are simply different forms of expres- 
sion of the same thing. The following notation will be 
•used : 



696 PLATE GIRDERS [Ch. XV. 

Z= length of span in feet; 
L\ = length of outside cover-plate in feet; 
L2 = length of second cover-plate in feet; 
A = total net flange area, square inches ; 
a\ =net area of outside cover-plate, square inches; 
a2 =net area of second cover-plate, square inches; 
as =net area of third cover-plate, square inches. 

It has already been seen that if a beam simply supported 
at each end be loaded uniformly throughout the span, the 
bending moment at any point will be represented by the 
vertical ordinate of a parabola whose vertex is over the 
centre of span while the end of each branch is at one end 
of the span. It is assumed that the greatest bending 
moments in the plate girder, already computed, vary by 
the same parabolic law. This is not quite true, but suf- 
ficiently near for ordinary purposes. 

Then, as will be shown in the next Article, 



r 7 pi 7- 7 /«l+«2. J 1 jai 



-\-a2-\-a2, 



A 

In this case / = 68 feet and A =45.3 square inches. 

ai=a2=az=g sq. ins. 

Making these numerical substitutions, there will result 
Li =30.7 feet; L2 =42.9 feet; L3 = 52.5 feet. These lengths 
are clearly the minimum permissible. In actual construc- 
tion it is desirable to have the end of the plate from i to 
1.5 feet further from the centre, making the total length of 
the plate 2 to 2.5 feet greater than the length computed 
above. This lengthening of the cover-plate is essential 
in order that the cover-plate metal may be taking stress 
at the point where the plate is computed to begin. Also 



Art. no.] 



THE DESIGN OF A PLATE GIRDER. 



697 



as will be seen a little further on, the pitch of rivets in 
these ends of the cover-plates is made less than in the 
body of the plate for greater effectiveness where the plate 
begins to take its stress. The lengths of cover-plates 
then, beginning with the shortest, will be 33.2, 45.4, and 
5 5, feet. 

Another method of procedure, more accurate than the 
preceding, is to draw a moment curve on the effective 
span, which can readily be done by laying down as vertical 
ordinates the resultant or total moments as given in Table I. 
These moment ordinates would be 5 feet apart except 
at the centre of span. The lengths of cover-plates must 
be such as to give resisting moments of the flange stresses 
at least equal to the external bending moments shown 
on such a diagram. The moments of the flange stresses 
will require the centres of gravity of parts of the flange 
sections to be computed at each moment point. The 
following tabulation shows the elements of this method 
of procedure for the centre section of the girder: 



Section. 

One-eighth web plus flange angles 

First cover-plate 

Second cover-plate 

Top cover-plate 



Sq. Ins. 



18.3 

9 
9 
9 



Stress per 
Sq. In. 



16,000 
16,000 
16,000 
16,000 



Lever Ai 
Ft. 



6.41 
6.77 
6.83 
6.9 



Moment. 
Ft.-lbs. 



1,875,000 
976,000 
984,000 
994,000 



This operation must be repeated at each m^oment section 
of the girder, but the numerical work need not be repeated 
here, being precisely like that for the centre section. 

The net lengths of plates found by this method are 32.8, 
42.9 and 53.8 feet, a substantial agreement with the lengths 
found by the shorter procedure. 

In the compression flange the cover-plate lying on the 
angles should run the entire length of the girder, especially 



698 PLATE GIRDERS. [Ch. XV. 

if the girder be of the deck type, i.e., with ties resting upon 
the upper flange. That flange being under compression, 
it is advisable that the horizontal legs of the angles be 
supported throughout their entire length by riveting 
them to a cover-plate. This will add to the stiffness and 
carrying capacity of the flange. If ties rest directly upon 
the upper flange, their deflection tends to bend one side 
of it out of its horizontal position, but this tendency will 
be materially lessened by the added stiffness gained in 
riveting the horizontal angle legs of the flange to the cover- 
plate. 

Although this process of design has been used in con- 
nection with the tension flange, under the specifications 
the compression flange is to be made like the tension flange, 
i.e., a duplicate of it. 



Pitch oj Rivets in Flanges. 

Arts. 5 and 31 of the specifications relate to the rivets 
required to join the vertical legs of the flange angles to the 
web plate. Art. 31 requires that "The flanges of plate 
girders shall be connected to the web with a sufficient 
number of rivets to transfer the total shear at any point 
in a distance equal to the effective depth of the girder at 
that point combined with any load that is applied directly 
on the flange. The wheel loads where the ties rest on the 
flanges shall be assumed to be distributed over three ties." 

The chief function of these rivets is to transfer hori- 
zontal shear from the web plate to the flanges, as it is in 
this way that the flanges receive their stresses. If the 
rivets take the direct load of the locomotive driving wheels, 
as in the case of a deck girder like that being designed, 
they must resist the resultant stress due to both vertical 
and horizontal loads. 



Art. no.] THE DESIGN OF A PLATE GIRDER. 699 

Strictly speaking the number of rivets required between 
two moment sections, as shown in Fig. i, should be just 
sufficient to give the increase of flange stress in passing 
from one section to the next one toward the centre of span. 
Art. 31 of the specifications, therefore, requires more 
rivets than are needed except at the end of the span. It 
is always necessary, however, to introduce more rivets 
near the centre of span than is required by actual computa- 
tions, for the general stiffness of the girder. Indeed even 
more rivets are generally provided than those prescribed 
in Art. 31 of the specifications. 

If d is the effective depth of the girder at the end of 
the span and if the end shear or reaction is R, and if tA is 
the flange stress at the distance d from the end of span, 
then will the following equation of moments be found, 
neglecting the negative moment of any load within the 
distance d from the end of the span: 

Rd = tAd, 
Hence 

R=tA, 

This shows that an amount of stress equal to the end 
shear must be given to each flange within the distance d 
from the end. The number of rivets required by this 
computation is a little more than necessary if any load 
rests upon the girder between the end and the section at 
the distance d from it. It will be clear that the general 
provision of Art. 31, quoted above, is based upon this 
end shear requirement, and it is analytically incorrect, 
but the excess of rivets which it calls for adds to the general 
stiffness and capacity of the girder. 

The weight of one driving wheel is 30,000 pounds, and 
it is to be distributed over three ties or 42 inches. As 



700 



PLATE GIRDERS. 



[Ch. XV. 



the prescribed impact is loo per cent., the vertical load 
per horizontal inch of girder will be: 

Tr 2 X 3.0,000 1, 

V = ^-^ = 1430 lbs. 

42 ^ 

It is obvious that the flange stress taken by one-eighth 
of the sectional area of the web is received directly by the 
latter and does not affect the rivets through the vertical 
legs of the flange angles. If Ai is the actual net flange 
section of cover-plates and angles and A 2 the total flange 
area, including one-eighth of the web section, and if 5 
is the total shear at any moment section, while d is the 
effective depth of the girder, then the horizontal flange 
stress H to be taken up per linear inch by the rivets 

between two sections the distance d apart will be H =--r^. 

d A2 

The values of Ai and A 2, beginning at the end section 

of the girder, are as follows : 



Section 


A 


A 


End 


22.88 sq.ins. 


27 .26 sq.ins 


sft. 


22.88 '' 


27.26 " 


10 " 


22.88 '' 


27.26 '' 


15 " 


31.88 '' 


36.26 '' 


25 " 


40 . 88 ' [ 


45.26 '' 


Centre 


40.88 '' 


45.26 '' 



The unit (inch) increments H of horizontal flange 
stress found for the various sections by the preceding 
formula are: 



End H 



5 ft. 



10 



319,500^22, 



80.5 27.26 



H = 
H = 



= 3330 lbs. 
= 2920 ^' 

=2510 " 



Art. no.] THE DESIGN OF A PLATE GIRDER, 701 



15 ft. H = 


= 2170 lbs 


25 '' H = 


= 1560 '' 


Centre H = 


= 954 " 



Each of the above results gives the horizontal stress H 
in pounds per linear inch, over each 80.5 inches of girder 
flange for each moment section and to be taken up by the 
rivets. 

The rivet pitch p at any section will then be determined 
by the following formula if K is the working value of one 
rivet in shear or bearing: 



PVV^-\-H^=K. 

Each rivet bears against the web plate as well as against 
each vertical leg of the flange angle, and as the web plate 
is much thinner than the sum of the thickness of the two 
angle legs, the bearing value against the web plate will 
be much less than that against the angle legs. Furthermore 
each rivet is subjected to double shear, the two shearing 
sections of the rivets coinciding with the two faces of the 
web plate. K, therefore, must be taken as the least of the 
double shearing value and the bearing value against the 
web plate. The rivets to be used are |-inch diameter 
before being driven and the bearing value of such a rivet 
against a ^^-inch plate at 24,000 pounds per square inch 
is 9190 pounds and 14,430 pounds in double shear at 12,000 
pounds per square inch, both of these working stresses 
being in accord with the specifications. 

Applying the numerical results thus established to the 
formula for the pitch, 



there will result : 



702 PLATE GIRDERS. [Ch. XV. 

At end p=-^= ^ ^ r =^2.55 ins. 

Vi43o2+332o^ 

5 ft. point p = 9i9 ^_^ ^^ 83 '' 

V 1430^ + 2920^ 

10 " p= =3.18 '' 

15 " P- =3-53 " 



C i 



25 P= =4-34 

Centre ^= =6.26 •' 

If desired a curve can be drawn at the various points 
with the corresponding pitch as a vertical ordinate at each 
point. Such a curve will give the rivet pitch at g^ny point 
in the span, but such detail is not usually required. The 
above values of the pitch may be used, with judgment, 
without further computations for any part of the girder. 
Fig. I shows the pitch used at the different girder points; 
it is frequently adjusted to the position of the intermediate 
stiff eners. 

Pitch of Rivets in Cover-plates. 

The number 0/ rivets required in a cover-plate is at once 
determined from its net section. In the present case the 
net section of each cover-plate is 9 square inches, which, 
at 16,000 pounds, gives 144,000 pounds as the stress value 
of the plate. The rivets in the cover-plates are subjected 
to single shear and the single-shear value of one J-inch 
rivet is 7220 pounds. Hence the number of rivets required 

to develop the full value of one cover-plate is -^^ = 20 

7220 

rivets. Between the end of the cover-plate, therefore, 

and the point at which the next cover-plate outside of it 

begins, there must be at least 20 rivets. As a matter of 

fact considerably more than that number will be found, 



Art. no.] THE DESIGN OF A PLATE GIRDER. 703 

as the pitch must not exceed 6 inches in any case and it 
should not be more than 3 inches for a distance of 12 to 
18 inches from the end of the plate. It will be seen upon 
examining the drawing that these conditions are fulfilled. 

Top Flange. 

As this flange is in compression, gross areas may be 
used. If the provisions of Art. 30 and other Articles of 
the specifications be scrutinized, it will be found that they 
are fulfilled by the compression flange made up as shown 
in the figures, and they need no further detailed attention. 

End Stiffeners. 

The end stiffeners must be heavy members of their 
class and rigidly riveted to the girder, as they take the severe 
impact or pounding at the points of support due to rapidly 
moving heavy locomotives and trains. Art. 79 of the 
specifications provides that " There shall be web stiffeners 
generally in pairs, over bearings, at points of concentrated 
loading, and at other points where the thickness of the web 
is less than one-sixtieth of the unsupported distance between 
flange angles. . . . The stiffeners at the ends and at points of 
concentrated loads shall be proportioned by the formula 
of paragraph 16, the effective length being assumed as 
one-half the depth of girders. ..." This provision 
makes it necessary to treat the end stiffeners as a column, 
the working stress to be: 

^ = 16,000 — 70—. 

The column load in this case is the maximum end shear 
including impact allowance as given by Table II, i.e., 
319,000 pounds. 



704 PLATE GIRDERS. [Ch. XV. 

If two pairs of sXslxH-inch angles be assumed for 
trial with the 3|-inch legs against the web plate, remem- 
bering that they will be separated by the thickness of the 
plate, the radius of gyration of their combined section 
about an axis lying in the centre of a horizontal web section 
and parallel to the web will be 3.13 inches. The length 

o ^ 

of the column is — ^=40.25 inches =/. Hence the pre- 
scribed formula will give a working stress of 15,100 pounds 
per square inch. On this basis 

. . J siQ,ooo 

Area required = '^—^ = 2 1 sq.ms. 

15,000 

The actual sectional area of four of the assumed angles 
will be 23.24 square inches, which is sufficiently close to 
the area required to be accepted as satisfactory. 

The entire load is carried to the end stiffeners by the 
I -inch rivets which bind them to the web plate. The rivets 
are in double shear and bear on the web plate. It has 
already been seen that the bearing value on the wxb plate, 
9190 pounds per rivet, is much less than the double shear 

value. Hence the number of rivets required is ^—^ =35 

9190 

rivets. This computed number of rivets distributed 

throughout the length of the 3|-inch angle legs would make 

the pitch too great. The pitch should not exceed about 

4 inches, which would make the number of rivets about 

40. It is essential, as already indicated, that the end 

stiff eners be made exceptionally stiff and rigid. 

End stiff eners are not bent, but are riveted onto filling 

plates having the same thickness as the flange angle legs. 

These filling plates enhance the stiffness and resisting 

capacity of the end stiff eners as they, in fact, form a part 

of the latter. 



Art. no.] THE DESIGN OF A PLATE GIRDER. 705 

Intermediate Stiff eners. 

By referring to Art. 79 of the specifications there will 
be found an empirical formula giving the maximum dis- 
tance between intermediate stiffeners, providing, however, 
that that distance in no case shall exceed the clear depth of 
the web. Intermediate stiffeners are sometimes regarded 
as being equivalent to the vertical compression members 
of a Pratt truss, but as a matter of fact there is no rational 
system of basing their design on computations. They 
are almost invariably made of angles, but sectional areas 
are determined by experience. Inasmuch as the total 
transverse shear at the centre of span is small, they are 
sometimes omitted there. As a rule they are never placed 
farther apart than the depth of web plate. 

As this girder is to carry a heavy railroad load pre- 
sumably at high speed, 5X3^Xf-inch steel angles will 
be used with the 3 J inch leg placed against the web 
plate. As the transverse shear increases toward the end 
of the span, the distance apart of these intermediate 
stiffeners will correspondingly be decreased. In the central 
part of the span this distance is seen to be 5 feet if inches, 
but near the ends it is reduced to 3 feet 5 J inches. The 
pitch of the rivets in these intermediate stiffeners may vary 
from 3 inches to 5 or 6 inches, the greater pitch being near 
the mid depth of the web. 

Splices in Flanges. 

It will be found that cover-plates and flange angles 
may be purchased of full lengths required on this plate 
girder. When, in general, the girders are so long as to 
require splicing of the parts of flanges, those joints for the 
tension flange must be so designed as to leave the net 
section as large as practicable, as the entire stress must be 



7o6 PLATE GIRDERS. [Ch. XV. 

carried by the net section. It is good practice and cus- 
tomary not to have two joints in adjacent parts concur, 
i.e., there should be breaking of joints so as to have a 
joint in one part only of the flange at the same section. 
In this manner the net section at each joint may attain 
its maximum value. In the splicing of angles both legs 
should be spliced. In compression, riveted joints can 
scarcely be expected to transfer stresses by abutting sur- 
faces in those joints. They should be spliced about as 
effectively as tension joints, although the question of net 
section does not arise, the gross section being available. 

Splices in Web Plates. 

As one-eighth of the gross web-plate section is considered 
as resisting bending as a part of the flange area, the rivets 
at a web-plate splice must be suflicient to resist the cor- 
responding bending moment. This web-plate moment is, 
therefore, 

16,000^, 7 >.yO 9 r • 11 

• — - — X-tX8o.5^ = 5,670,000 m.-lbs. 
8 , 10 

There must be two splice- plates, one on each side of 
the web, each of which need not be as thick as the main 
plate, but in this case f-incli splice-plates have been used 
so that the intermediate stiff ener need not be bent. For 
this size of girder there should be three rows of rivets on 
each side of the joints. If it be assumed that the pitch 
be 4 inches in each row, there will be nine rivets in each of 
the three rows between the mid depth of the web and the 
back of the flange angles. If the loads carried by these 
rivets in resisting bending vary directly as the distance 
from the neutral axis at mid depth, their resultant will 
act at 1X40 = 26.7 inches from that line. The bearing 



Art. iio.l THE DESIGN OF A PLATE GIRDER. 707 

value of a |-inch rivet against the i^-inch web is 9190 
pounds. Hence the resisting moment of the 54 rivets 
on one side of the joint is: 

M = ^^^-^X2 X26. 7 =6,600,000 in.-lbs. 



As this is greater than 5,670,000 in. -lbs., the proposed 
arrangement of the joint is satisfactory. The two splice- 
plates will, therefore, each be 19 X f inches by 5 feet 8| inches, 
as shown in Fig. i. 

In general every joint splicing should be tested for 
the transverse shear which it must carry. In this instance 
it is clear that the splice-plates will carry more shear than 
the web. 

General Considerations. 

The girder proper with its flanges, web, and stiff eners 
has been designed in this article without indicating the 
manner of connecting such lateral or cross bracing as 
would be required in the complete design of a railroad 
plate-girder span. The design of such bracing would be 
supplementary to the actual design of the girder as made, 
and it is the purpose here to illustrate only those principles 
belonging to the design of the girder proper. The design 
of the bracing and the details of its connection with the 
girder belong rather to bridge construction than to the 
subject treated here. Fig. 2 has been introduced, however, 
as an illustration to indicate the general features of the 
complete structure. 

Large plate girders are not always built complete in 
the shop, although girders nearly 100 feet in length are 
frequently and perhaps usually so completed at the present 
time. When it is necessary to build them in portions 



7o8 PLATE GIRDERS. [Ch. XV. 

and rivet the portions together in the field, the general 
principles governing the construction of the necessary 
field-joints are precisely the same as those illustrated in 
this article. They are simply adjusted or adapted to the 
exigencies of each particular case. 

The bill of material and estimated weight of a single 
girder as designed is as follows : 

Pounds. 

Two 80" Xi^" web plates, 21' \i\" long 5,236 

One 80" Xi^" web plate, 26' |" long ; . . 3,094 

Four 6"X6"Xf" angles, 70' long 8,036 

One 14" X I" cover-plate, 70' long 2,499 

One 14" Xf" cover-plate, 55' 5I" long 1,981 

Two 14" Xi" cover-plates, 47' 6^" long 1,696 

Two 14" X I" cover- plates, 33' 3" long 1,190 

Eight ^"XzV'XW angles, 6' 7" long 1,037 

Twenty-eight 5" X3I" X I" angles, 6' 7" long 1,917 

Four 10" Xf" filler-plates, 5' 8|" long 581 

, Four 19" Xf" splice-plates, 5' 8|" long 1,106 

Twenty-four 3^"Xf " filler-plates, 5' 8|" long 1,222 

Two 14" XI" sole-plates, i' 6" long 107 

Rivets 800 

Total for one girder 30,502 

The weight of girder per linear foot therefore is: 

^-^ — =436 lbs. 
70 

If the plate girder were of the through type, there 
would be no change whatever in the procedures of design 
which have been followed, but in order to give a better 
appearance to the ends they would be formed as shown in 
Fig. 5. The latter figure shows the same end stiffness, 
depth of girder and the same flange angles as Fig. i. 

Art. III. — ^Length of Cover-plates. 

There are various methods of determining the lengths 
of cover-plates of plate girders involving simple compu- 



Art. III.] LENGTH OF COVFM-P LATHS. 709 

tations only, which are well illustrated by the following 
procedures : 

The first of these procedures is based on the assump- 
tion that the depth of the girder is uniform and that the 
bending moment throughout the length of girder varies 
as the ordinate of a parabola as in the case of uniform 
loading. The following notation is required: 

/ = effective length of span either in feet or inches ; 
L= length of cover-plate required in the same unit as /; 
A = total net flange area; 
a = net cover-plate area required. 

Since the flange and cover-plate areas vary directly 
as the flange stresses, and as the latter vary as the ordi- 
nates of a parabola when the depth of girder is constant, 
the following equation will result: 

P~A' 



or 

, a 

(i) 



='^g 



A 

Eq. (i) will give the length of the cover-plate whose 
area of section is a. Any convenient unit may be taken 
for a and A , but the square inch is ordinarily employed. 

If there are several cover-plates, a is to be taken suc- 
cessively the area of the first, second, third, etc., cover- 
plates in summation, i.e., it will first be taken as the net 
sectional area of the top cover, then as the net sectional 
area of the top cover added to that of the cover-plate 
below it, and so on. 

The second method is the following, and is applicable 



7IO PLATE GIRDERS. [Ch. XV. 

to the case of a girder with varying depth, the notation 
being as follows : 

Let Z£;= uniform load per linear foot, or "equivalent 

uniform load" per linear foot; 

d and d' represent the effective depths of girder in 

feet at the centre of span and at the end of 

the cover-plate respectively; 

A — a = a'= area of flange section at the end of 

cover-plate ; 
r = permissible flange stress per square inch; 

the bending moment at the end of the cover-plate will 
then be 

M=w^-~[~] =AdT-w^=d'a'T. . . (2) 
8 2 \2 / 8 ^ ^ 

By solving the second and third members of the pre- 
ceding equation there will result 



L = 2\/\ 



{Ad-a'd')T ^ ^^ [{Ad-a'd')T 



^■83\^ " :^ . . (3) 



w ^ -^ w 



It must be remembered that the application of either 
of the two preceding methods will give the net length of 
the cover-plate. There must be added 12 to 18 ins. at 
each end with rivets closely pitched so that the cover- 
plate may certainly take its stress at the points where its 
effectiveness should begin. 

Art. 112. — Pitch of Rivets. 

A simple method of finding the pitch of rivets piercing 
the vertical legs of the flange angles and the web plate of a 



Art. 112.] PITCH OF RIVETS. 711 

plate girder at any section of the beam may readily be found 
by using the general but elementary expression for the bend- 
ing moment, 

IPx=M. 

By differentiating this equation, 

IP.dx = Sdx=dM; 

.:> representing the total transverse shear. 

If dM is the change of bending mom.ent for the distance 
along the flange represented by the pitch of rivets, p, the 
change of flange stress for the same distance will be found 
by dividing dM by the effective depth of the girder, d. If 
the pitch of rivets, p, be placed in the preceding equation in 
place of dx, the corresponding change of flange stress will 
represent the amount of stress transferred to the flange by 
one rivet. Representing that variation of flange stress by 
V, the last of the preceding equations may be written 

Sp=dv; :. P=^' : 

In this equation v represents either the bearing capacity 
of one rivet against the web plate or against the two flange 
angles, or the double shearing value of the same rivet, i.e., 
the least of those three values. Ordinarily the bearing of 
the rivet against the web plate will be less than either of the 
two other quantities; hence that bearing value would then 
be substituted for v. In general the least of the three pre- 
ceding values for one rivet is to be substituted for ^' in an 
actual computation. The total transverse shear 5 is always 
known at any section or may readily be determined. The 
preceding formula for the pitch, therefore, is a very simple 
one and is much employed. .__„__. 



CHAPTER XVI. 

MISCELLANEOUS SUBJECTS. 

Art. 113. — Curved Beams in Flexure. 

If beams are sharply curved, i.e., if the radius of curva- 
ture of the neutral surface is comparatively small, the for- 
mulae expressing the common theory of flexure for such 
beams will contain the radius of curvature and corre- 
sponding variations from the formulae for straight beams. 

Let Fig. I represent part of a curved beam subjected to 
flexure, AC representing the radius of curvature at the 




Fig. 



point A before flexure while CA' represents the radius 
of curvature of the same surface after flexure takes place. 
OAO^ represents the neutral surface. A^f is the continu- 
ation of C'A\ Similarly A'b is the continuation of CA'. 
Finally, A'b' is drawn parallel to CA. def represents the 
normal section of the beam and AA^ is supposed to be 
a differential of the length of the neutral surface. 

712 



Art. 113.] CURBED BE^MS IN FLEXURE. 713 

The ordinate dzy is measured from A as an origin 
toward B or D, respectively, z is the varying width of 
the normal section of the beam and hence it is measured 
normal to y and x, the latter being measured along 0A0\ 
A differential of the section of the beam is zdy. 

As the normal sections of the beam are assumed to 
remain plane after flexure, let the rate of strain, i.e., the 
strain per unit of length of fibre at any point distant y 
from the neutral surface be uy, u being the apparent 
rate of strain at unit distance from the neutral surface. 

By referring to Fig. i there may at once be written: 

■ b'b=da^; hh" = (dx-^do^uyi. 
By similarity of triangles, 

do(^+dx{i^-'^' 



'"^ dx 



(i) 



This equation gives at once : 



r 
r-r' 7"' , , 

U=-, r-, = (2) 

{r-^yy r+y ^ ^ 

If the beam were originally straight, in which case the 
radius of curvature r = 00 , eq. (2) would take the form 

u=-, the usual expression for the rate of strain at unit 

distance from the neutral surface of a straight beam. 
If again the radius of curvature is sufficiently large, so that 
r may be written for r-\-y without sensible error: 

"=/-7- ^3) 



714 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

This expression for m may be used for curved beams if 
the curvature is not too sharp. 

If the radius r is infinitely great, u = -, which is the 

value for a straight beam. 

Eq. (2) shows that the rate of strain u at unit distance 
from the neutral surface and corresponding to the rate of 
strain at any distance y is variable, as y appears in the 
denominator in such a way as to make u smaller the 
greater the distance of the fibre from the neutral surface. 
This is in consequence of the curvature of the beam and 
results from the assumption that normal sections plane 
before flexure remain plane after flexure. With the 
increase of length of fibre due to curvature as its distance 
from the neutral axis increases, a less rate of strain is 
required to keep the section plane after flexure. This 
assumption is not strictly true, and it may be a matter of 
doubt whether it is necessary or advisable even in the 
interests of correct analysis. 

If k is the fibre stress of tension or compression at any 
distance y from the neutral axis, there may be at once 
written : 

k=Euy=E(^-,-ijy^^ (4) 

The stress on an element zdy of the section will then be : 

«,=£(^,^.)sg (s) 

Let k' and k'^ he the intensities of stress at the distances 
y' and -y" from the neutral surface. Then by eq. (4): 

k' ^ y' r-y" 
. k" r+y ~y"' 



Art. 113.] CURBED BEAMS IN FLEXURE. 715 

From this equation: 

liy'=y", eq. (5a) becomes: 

k" = -k''-±^, (5fa) 

r —y 

Eq. (55) shows that the intensity of stress at a given 
distance from the neutral axis will be greater on the concave 
side of the curve than on the convex, and that this relation 
holds until the radius of curvature becomes infinitely 
great. 

In order to locate the neutral axis the integral of the 
two members of eq. (5) between the limits of y and —y 
must be placed equal to zero, giving eq. (6) : 



X>==^&-C 



^^=0. ... (6) 



Again, the bending moment formed by the direct 
stresses of tension and compression in the section may be 
written in the usual manner as follows, M representing the 
moment : 



-&-)x: 



r-\-y 

^=^ir,-)i f5f (7) 



Eq. (6) shows that the neutral axis will not pass through 
the centre of gravity of the section. As the intensity of 
stress on the convex side of the curve will be less than if 
the beam were straight, the neutral axis will be on that 



7i6 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



side of the centre of gravity of the section toward the con- 
cave surface of the beam. Eq. (7) shows, again, that the 
integral is not the moment of inertia of the section about 
the neutral axis, but it will reduce to that if the radius 
of curvature r be supposed infinitely great. 

The integrations shown in the second members of eqs. 
(6) and (7) can at once be made when the form of cross- 
section is known. Inasmuch as this analysis for curved 
beams finds one of its important applications in connection 
with the design and carrying capacity of large hooks, a trape- 
zoidal cross-section shown in Fig. 2 will be assumed by 

way of illustration, and from 
that the rectangle section at 
once results. In that figure 
the larger end CD of the trap- 
ezoid will be considered to 
lie in the concave or inner 
surface of the hook and at 
right angles to the plane of 
the hook. As the trapezoid 
is symmetrical, a = \FH, and 
the angle of inclination of a 
sloping side as HD to the 
centre line will be taken as a. 
Then z will represent one-half of the width of the trapezoid 
at any point: 




Fig. 2. 



z=a + {yi —y) tan a. 



(8) 



If z be inserted in eqs. (6) and (7) there will be required 
the following integrations in which yi-\-yo=d: 



/- 



'' ydy _^ 

yo^+y 



rlog 



r+yi 
r-yo 



. (9) 



/: 



Art. 113.] CURBED BEAMS IN FLEXURE. 717 

r'^^=d(yi^-r)+rUog'±y-\ . . do) 
J-y,r+y \ 2 / r-yo 

" t^ =rH-hrd{y, -yo) +id(d' -syiyo) 

-^,og(^;). . . („) 

If these values of z and the integrals given in eqs. (9), 
(10) and (11) be substituted in eq. (6), there will at once 
result : 

log'-±yi= l^ '^ I. . . . (12) 

r — yo r\ ir +yi) tan a -\-a] 

As known quantities let r-\-yi=R and r—yo=Ro, 
then eq. (12) may take the form: 

r log -^(R tan a -\-a) = rd tan a -\-d[ - tan a +a ) . 

Kq \2 / 

Hence : 

J(-tana+a) 

(r log — -d\ tan a+a log — 

After r is determined by eq. (13) there w^ll at once 
result : 

yi=R-r and yQ=d-yi. . . . (14) 

If the section is rectangular, Q:=tan a=o, hence, 

^ = ^ and yi^R- — — . . . (14a) 

log^ logf- 



7i8 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

If the section is triangular, a=o and the second member 
of eq. (13) will be correspondingly simplified as follows: 

r=-i -5 \ (15) 



(i?log|-.) 



As this expression is independent of o:, yi and yo remain 
unchanged whatever may be the value of that angle. 

Having thus found yi and >'o, the position of the neutral 
axis of the section is determined and the expression for 
the bending moment can now be written by the aid of 
eqs. (4) and (7), 'the latter being the general expression 
for the bending moment. By the aid of eq. (4) the in- 
tensity of stress in the extreme fibre at the distance yo 
from the neutral axis may be written as follows : 

ybo=-£(-,-i)-^ (16) 

\r Jr-yQ 

Hence, 

4.-) = -^- • ; • • <■" 

By introducing the second member of eq. (17) in eq. 
(7) as w^ell as the value of z from eq. (8) and the integrals 
given in eqs. (10) and (11), the following value of the 
moment M will result : 

-M=^M!:^:^P'((a+3;,tan«)^^-tana^l , (i8) 



I \ \U, -r/1 tdii UI.J ; tail ex ; > 

yo J-yoi r+y r+y] 

2ko{r-yo){.^.^^,.^_ .(d 



yo 



I (a + (r -^yi) tan a) (-{yi -yo) 



.dr+rnog'-±y^)y-^^{d^-iyryA . . (19) 

r-yo/ 3 J 



Art. 113.] CURl^ED BEAMS IN FLEXURE. 719 

As is evident, the factor 2 appears in the second members 
of eqs. (18) and (19), for the reason that the section taken 
is symmetrical and the varying ordinate z is half the width 
of section at any point. If a were taken as the extreme 
width of section on the narrow side instead of half that 
width and if a were to be so taken that (yi—y) tan a 
added to a represents the full width of the section at the 
point located by y, the factor 2 would be omitted from the 
second member of the value for M. 

If the section is rectangular a = tan a=o and the 
expression for the moment M then becomes : 

_M ^ ^feo(r-yo) f p(^^ _^^) _^^^^, 1 r±n\] (^^^ 
yo i \2 r-yo/ J 

If the section were triangular a = o in the second member 
of eq. (19). 

These equations may be employed in the design of curved 
beams of any form of cross-section or degree of curvature 
when those based on the common theory of flexure for 
straight beams are not applicable. As a general statement 
it may be said that the formulas for straight beams may be 
used without essential error in all cases except those of such 
special character as hooks and other structural or machine 
members in which the curvature is sharp. The applica- 
tion of the preceding formula to the case of hooks will be 
illustrated in the next article. 

Art. 114. — Stresses in Hooks. 

The diagram of a hook shown in Fig. i illustrates the 
conditions of loading to which hooks in general are sub- 
jected. The material to the right of the point of applica- 
tion of the load is subjected to no stress whatever except 
in a secondary way near that point. On the left of the 



720 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



load, however, the arc of the hook, supposed to be circular 
in this case, is subjected to direct stress, shear and bending, 
the bending moment increasing as that part of the hook 




parallel to the loading is approached, but it decreases in 
passing on to the shaft of the hook supposed to be in line 
with the load. The section of miaximum bending AB 
is subjected to the combined direct pull of the load and 



Art. 114. 



STRESSES IN HOOKS. 



721 



the bending moment equal to the load multiplied by the 
normal distance from its line of action to the centre of 
gravity of the section. This cross-section of greatest 
bending moment will first be treated as if subjected to., 
pure flexure. The necessary simple analysis required to 
determine the greatest intensity of stress in the section will 
then be made. In the section 
of greatest bending moment 
there is no shear. 

The cross-section of the 
main part of a hook maybe 
taken as .approximately trap- 
ezoidal, as shown in Figs, i 
and 2. In the present in- 
stance the greatest dimension 
of this cross-section lying in 
the central plane of the hook 
will be taken as 5 inches and 
the corners will be rounded 
approximately as shown. 

Obviously the integrations 
of eqs. (9), (10) and (11) of the 
preceding article do not rep- 
resent accurately the approx- 
imate trapezoid of Fig. 2 . This 
integration or its equivalent, 

however, may be accomplished with sufficient accuracy 
by a number of approximate processes, i.e., by transformed 
figures and by dividing the section into a sufficient 
number of small parts. A simpler method and one giving 
reasonably accurate results is to draw two lines F^C and 
FD in such a way as to make a true trapezoid whose resist- 
ing moment will be essentially the same as the approx- 
imate trapezoid. This will be accomplished if the two 




X 2V4- > 



Fig. 2. 



722 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

lines indicated be drawn in such a way that each area 
between a broken line as F'C and the inclined full line 
of the actual section be three times the combined area 
between CB and the curved end of the section and between 
AF' and the other curved end of the section. This rela- 
tion results from the fact that the bending stress between 
the tw^o lines indicated varies in intensity from zero at the 
neutral axis to nearly the maximum in the extreme fibre 
of the section and has its centre at two-thirds of the distance 
from the neutral axis to the extreme fibre. The relation 
indicated, therefore, makes the bending moments of the two 
parts inside and outside of the actual section equal. This 
construction will give for one-half the modified figure: 

AF' =AF = .^^ inch=a; BC = 1.1 inches; a =S° 30'; 
tan a' = .148; i? = 5+2.2 =7.2 inches; R = 2.2 inches. 
d = s inches. 

Fig. I shows that R and Rq are the interior and exterior 
radii respectively of the arc of the hook where the section 
of greatest bending moment exists. By introducing these 
numerical quantities in eq. (13) of the preceding article 
there will at once result : 

r =3.87 inches. 
Hence, 

yi =R-r = s-33 inches; 

yo=d—yi=i.6j inches. 

By inserting the same numerical values together with 
yi and yo in eq. (19) of the preceding articles, the value of 
the bending moment becomes : 

M =4. 88^0 (i) 



Art. 114.] STRESSES IN HOOKS. 723 

This moment obviously can be expressed in terms of the 
intensity of stress in the extreme fibres on the opposite 
side of the section, i.e., 3.33 inches from the neutral surface. 
By eq. (4) of the preceding article: 



ki =ko 



(r-yo)yi 
yoir+yiY 



After substituting' the values of the quantities already 
determined there will be found ^i=.6i/^o. Or there may 
be written from the same eq. ^0 = 1.64^1. The bending 
moment expressed in terms of the greatest intensity of 
stress in the extreme fibres is obviously the form desired 
for practical purposes. 

Let the hook shown in Fig. i be supposed to carry 
a load of 20,000 pounds. The centre of gravity G of 
the actual cross-section is 2.13 inches from the side CD 
of the cross-section. Fig. 2. Hence the load assumed 
will cause a bending moment about the line GH equal 
to 20,000 X(2. 13 -1-2.2 =4.33) =86,600 inch-pounds. It is 
to be observed that inasmuch as the 20,000 pounds is taken 
as uniformly distributed over the cross-section the lever 
arm of the load is the normal distance from its line of 
action to the centre of gravity of that section, although the 
resisting moment of internal stresses has the axis deter- 
mined by eq. (14) of the preceding article, the two axes 
being parallel to each other. 

The greatest intensity of tensile bending stress in the 
section therefore takes the following value: 

, 86,600 11 . , X 

^0 = 5^- = 17^40 lbs. per sq.m. . . (2) 

4.00 

The uniformly distributed tensile stress equal to the 
load will act upon the entire actual area of section, which 



724 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

is 7.9 square inches. Hence, that tensile intensity will 

be — '■ =2530 pounds per square inch. The resultant 

7-9 
greatest intensity of stress in the entire section will be: 

17,740 + 2530 =20,270 lbs. per sq. in. . . (3) 

The resultant intensity on the opposite side of the sec- 
tion at A, Fig. 2, will be, since ki =.6iko\ 



— 17,740 X.61 +2530 = —8291 lbs. per sq.in. . 



(4) 



The minus sign is used because the bending stress is 
compression throughout that part of the section indicated 

It is commonly observed in actual experience that 
hooks or other similar bent members break at the inside 
of the section where the curvature is the sharpest. The 
eqs. (4) and {s^) of the preceding article indicate clearly 
the reason for such failures as the intensity of stress k in 
the extreme fibre is shown to vary inversely with the radius 
of curvature r-\-y. When, therefore, the curvature is 
sharp, i.e., the radius of curvature is small, the fibre stress 
k increases rapidly, especially on the inside of the curve 
where the radius of curvature is r—y. 

This example shows the general method of treating the 
stresses in hooks by the common theory of flexure based 
on the assumption that normal sections plane before flexure 
remain plane after bending. 

It is well known that this assumption is not strictly 
correct, and it is further known that the ordinary or com- 
mon theory of flexure is not accurately applicable to such 
short beams as are contemplated in the theory of hooks. 



Art. 114.] STRESSES IN HOOKS. 725 



Comparison with the Theory of Flexure for Straight Beams. 

It is indicated above that the assumptions on which the 
preceding analyses are based are not strictly correct. If 
it be assumed that the intensity of stress varies directly 
as the distance from a neutral axis passing through the 
centre of gravity of the section, as for straight beams, and 
if k\ is the greatest intensity of stress in the extreme 
fibres {FF', Fig. 2) the bending moment will be: 

M=^-^ (5) 

yc 

In this equation I is the moment of inertia about an axis 
through the centre of gravity G, Fig. 2, while yc is the 
distance of that axis from the most remote fibre at A. 
The moment of inertia I of the actual section shown in 
Fig. 2 about a neutral axis through the centre of gravity 
G Sit the distance 2.87 inches from A is 14.9. Hence, 
the bending moment on the preceding assumption is: 

M=^k\=s.2k\ (6) 

r 2 

As the fraction ^^=1.07 this assumption is seen 

4.88 

to give a result only 7 per cent, greater than that of the 
analysis for curved beams if the extreme fibre stress is the 
same in amount in both cases. It is true that the result 
has the apparent defect of placing the greatest intensity 
of stress on the wrong end of the section. 



Art. 115. — Eccentric Loading. 

The analysis of stresses produced in a column or other 
structural member by eccentric loading has already been 



726 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



discussed in preceding articles, but it is desirable to con- 
sider some further and more general features of that analysis. 
A column or structural member is said to be eccentric- 
ally loaded when it carries a force or load acting parallel 
to its axis but not along that axis. The perpendicular 




Fig. I. 



distance between the axis of the piece and the line of action 
of the load is called the eccentricity of the latter. 

Let Fig. I represent the normal cross-section of such a 
member when the load P acts at any point Q in that cross- 
section. The load P will then act parallel to the axis of 
the piece, but at the distance CQ from it, C being supposed 
to be the centre of gravity of the section. The ellipse 



Art. 115.] ECCENTRIC LOADING. 727 

drawn with C as its centre is the elHpse of inertia, the semi- 
axes ri and r2 being the principal radii of gyration of the 
normal section. Any semi-diameter as CQ' represents 
a radius of gyration r\ 

If the force or load P acts at any point whatever, as Q, 
and parallel to the axis of the piece, it will create a bending 
moment equal to PxQC. If x' and y^ are the coordinates 
of Q the components of that moment will be Px' and Py\ 
the former about the axis Y, the latter about the axis X. 
1 1 and 1 2 being the principal moments of inertia, as already 
indicated, the intensities of bending stresses produced by 
these two component moments at any point, whose co- 

Px' PV 

ordinates are x and y, will be —j—x and ----y, respectively. 

i 1 J-2 

Furthermore, the load P will produce a uniform normal 
stress over the entire cross-section of the member, if A 

p 
is the area of that cross-section, represented by — . The 

resultant intensity of stress k therefore at any point of the 
section will be : 

At the neutral axis the intensity of stress is equal to zero, 

hence, 

x'x y'y , s. 

— j+^=-i. ...... (2) 

Eq. (2) is the equation of a straight line, i.e., the neutral 
axis, along which the intensity of stress is zero, x and y 
being the variable coordinates. It is obvious from eq. (2) 
in connection with the general considerations respecting 



728 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

the action of the load P that the position of the neutral 
axis will depend upon the magnitude of that load and the 
distance of its line of action from the axis of the member. 
If X and y are zero, there will be no bending, and the section 
of the member will be subjected to uniform compression 
only. 

If the point of application Q of P is on the curve its coor- 
dinates x' and y' must satisfy the equation of the ellipse: 

^'2 r^f2 

^+2^ = 1 (3) 

The equation of a straight line tangent to the ellipse 
at a point whose coordinates are x' and y' is: 

"-^+2^ = 1 (4) 

When the point of application of P is on the ellipse, x' 
and y' have the same values in eqs. (2) and (4). Hence in 

that case -p- also has the same value in the two equations, 
ax 

showing that the neutral axis is parallel to the tangent to 

the ellipse at the point where P acts. If in eq. (4) —x' 

and —y' be substituted for -\-x and -\-y, that equation will 

become identical with eq. (2), i.e., for this case the neutral 

axis is tangent to the ellipse at a point diametrically opposite 

to the point of application of the load P ; in other words, 

the load is applied at one extremity of a diameter and the 

neutral axis is tangent to the curve at the other extremity 

of that diameter. 

In Fig. I if the load P is applied at Q' (on the curve) 
the neutral axis N'B' will be tangent to the ellipse at B' , 
the other extremity of the diameter Q'B'. 

If the point of application of the force P moves along 



Art. 115.] • ECCENTRIC LOADING. 729 

the indefinite straight line BQ, the coordinates x' and y' 

x' 
will vary in the same proportion, making — a constant. 



(s) 



From eq. (2) : 








dy 
dx 


x'r2^ 
y' ri2 



X 

Hence, as — is constant, all neutral axes will be parallel 

y 

while the point Q moves along a straight line. 

Again, the coordinates x and y of the points of inter- 
section of the line QB with the neutral axes must neces- 
sarily be opposite in sign from x' and y' , as the origin C 
lies between them. If therefore —x and —y be inserted 
in eq. (2) : 

^^+^^1 (6) 

By similarity of triangles, a being a constant: 

XX II I / \ 

— : = -=a .. XX =ayx=ayy. ... (7) 

y y y yy n// 

Eq. (7) in connection with eq. (6) shows that each of 
the quantities x'x and y'y is constant, and that is equivalent 
to making the products of the segments of the line QB on 
either side of C constant: 

QCy.CB=Q'C^CB' ^Y"'=Q"Cy.CB". . . (8) 

As the point of application of P will always be given, the 
quantity to be found will be the distance from the centre C 
to the neutral axis, which may be called v. The semi- 
diameter y' =CQ' at once becomes known after the ellipse 
of inertia is constructed. In general, therefore: 

y'2 

'=QC (5) 



730 MISCELL/INEOUS SUBJECTS. [Ch. XVI. 

In some cases the reverse problem is given, i.e., v is 
known and the distance of the point of application of the 
load P is required. Hence, 

■ QC=- do) 

V 

Rotation of the Neutral Axis about a Fixed Point in It. 
One feature of eq. (2) remains to be considered before 
the actual application of the preceding results can be made 
to form a complete graphical construction. If the co- 
ordinates X and y of the neutral axis be considered constant, 
while the coordinates x' and y' of the point of application 
of the load Pvary, eq. (2) shows that the path of the move- 
ment of the point of application of P will be a straight 
line, since the equation is of the first degree in respect 
to x' and y' . This is equivalent to a movement of rota- 
tion of the neutral axis about the fixed point whose coor- 
dinates are x and y, while x' and y' determine the path 
through which the line of action of P moves. The same 
result can be shown by treating eq. (i) in precisely the same 
manner for a fixed or constant value of k, that constant 
being zero for the neutral axis. 

The preceding procedures may be applied to a number 
of problems, one or two of which will be illustrated. It is 
sometimes desired to determine that part of the cross-sec- 
tion of a member of a structure, or sometimes of the struc- 
ture itself, within which a resultant load may be applied 
anywhere without any change in the kind of stress induced, 
usually compression. 

Application of Preceding Procedures to Z-har and Rectangular 

Sections. 

Let it be required to ascertain within what part of a 
Z-bar section an axial compressive force may be applied 
without any part of the section being subjected to tensile 



Art. 115.] ECCENTRIC LOADING. 731 

Stress. The Z-bar section is shown in Fig. 2, the depth 
of bar being 6 inches and the thickness of metal f inch. 
As this section is unsymmetrical the axes for the principal 
moments of inertia passing through the centre of gravity 
C of the section will be incHned to the central plane of 
the^ web. The ellipse of inertia MVNU has MN for its 
major axis and UV for its minor axis, the former representing 
a moment of inertia of 52 and the latter a minimum moment 
of inertia of 5.7, the corresponding radii of gyration being 
ri=2.55 inches and r2=.8i inch. 

If no part of the cross-section of the bar is to 'be sub- 
jected to tension, the outer Hmits or Hnes of that section 
such as TS, SO, OL, etc., may be neutral axes for different 
positions of the load^, but in no case must the neutral 
axis He in any part of the metal section, even to cut across 
a corner of it. This means that TS, SO, OL, LH, HE, 
and ET will be successively considered neutral axes.' Let 
ET be the first neutral axis considered or, rather, ET and 
OL may be considered concurrently, as they are parallel 
to each other and at the same distance from the centre of 
the ellipse. First draw tangents to the ellipse parallel to 
ET and OL as shown in the figure. The points of tangency 
will fix the diameter DA, which is then extended to R 
and W in the assumed neutral axes. As shown in the 
preceding demonstration, the square of half the diameter 
represented by AD will be equal to CR multipHed by CA 
the distance from the centre of the ellipse to the point 
of apphcation A of the force P. The distance CR is the 
V of eq. (10), while CA is the distance QC desired, / is 
half the diameter determined by the two points of tangency 
Dividing the square of half the diameter by CR locates 
the point A, one of the points desired. In precisely the 
same manner D is located by dividing the square of half 
the same diameter by CW = CR. 



732 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

Tangents to the ellipse parallel to TS and HL are then 
drawn as shown, one at A^, as indicated, at the lower 
extremity of the ellipse and the other at the upper extremity, 
thus locating the diameter NCF. Squaring half the diam- 
eter so determined and then dividing by the distance from 
the centre of the ellipse to TS or HL along the diameter 
NF, the points B and F are found. In a precisely similar 
way the vertical tangents indicated are drawn parallel to 
SO and EH, determining the corresponding diameter. By 
the use of that diameter in the manner already indicated, 
the points G and K are located. 

The points A and B are points of application of the force 
P when ET and TS respectively are neutral axes. In the 
preceding sections of this article it has been shown that if 
a neutral axis such as ET be revolved about a point in it, 
as T, to the position TS, the corresponding path of the point 
of application of the load will be a straight line, and in this 
case AB will be that straight line, since the two points A 
and B correspond to the neutral axes ET and TS. By 
similarly connecting the other points, the closed figure 
ABKDFG is found. So long as the force P acts within 
this area no part of the section can be subjected to tension, 
but if the point of application is outside of this figure 
some part of the section will be in tension 

The closed figure thus established is called the '' core 
section." Although it possesses much analytic interest, 
the ordinary operations of the engineer are such as to make 
it of comparatively little value in actual structural opera- 
tions. 

If any line such as Z'L parallel to a tangent to the 
ellipse at g be drawn through a corner L of the Z-bar sec- 
tion, and if a line dgZ' be drawn through the same point 
of tangency and the centre C, cutting the side of the core 
at d, it is shown in the preceding section of this article 



Art. 115.] 



ECCENTRIC LOADING. 



733 



that the product of dC by CZ' is equal to the square of the 
semi-diameter Cg of the elHpse of inertia For any other 
position of a Hne Z'L the same general observation holds, 
the line always being parallel to a tangent to the ellipse 




Fig. 2.* 

at a point through which is drawn the line extending 
through the centre and cutting the side of the core. 

Probably the most usual section to which the core 



* A number of construction lines shown in this figure are drawn for use 
in the next article. 



734 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



procedure may be applied is the simple rectangle. A 
masonry structure having such a horizontal section must 
be designed so that compression only may always be 
found in it. A simple diagram of pressures will show that 
the resultant force or load must act within the middle 
third of the section, but Fig. 3 shows the core procedure 
appHed to the same axis. AB \s the length of the section 
and BD is the width. AB is usually taken as one unit. 
The ellipse OLMN is drawn with its 
semi-major axis LC representing the 
greatest radius of gyration of the rec- 
tangle and the semi-minor axis OC is 
laid off equal to the least radius of 
gyration. Two lines drawn tangent 
to the ellipse at M and N parallel to 
BD and ED will determine the axes 
of the ellipse, in fact already known, 
then dividing the square of each semi- 
axis by the normal distance of C from 
BD and ED, respectively, the dis- 
tances CF and CK will be found, thus 
fixing the points F and K. The points 
H and G are found in precisely the same manner, using the 
sides AE and AB respectively. As already indicated, the 
distance of H from BD will be one-third oi AB, while K 
will be one-third oi BD from AB. 




Fig. 3. 



General Observations. 

The preceding results show that bending combined with 
uniform stress induced by a load normal to the section 
will prevent the neutral axis from passing through the centre 
of gravity of the cross-section. Furthermore, in this general 
case the neutral axis or neutral surface will not be at right 
angles to the plane containing the axis of the piece and the 



Art. ii6.] GENERAL FLEXURE TREATED BY CORE METHOD. 735 

line of action of the force unless that plane contains one 
of the principal axes of inertia. 

Manifestly the neutral axis for any section will be on 
the opposite side of the centre of gravity of that section 
from the force P. Eq. (8) shows that if the force acts at 
C, making QC equal zero, CB will be infinitely great, which 
means that the stress will be uniformly distributed, i.e., 
there will be no bending. On the other hand, if the force 
P is at an indefinitely great distance from C, making QC 
infinity, then will CB be equal to zero, i.e., the neutral 
axis will pass through the centre of gravity. This is the 
ordinary case of flexure and it is equivalent to taking all 
load on the member at right angles to its axis. 

Art. 116. — General Flexure Treated by the Core Method. 

The procedures given in the preceding article may be 
used for the general problem of flexure for straight beams 
of any form of cross-section carrying any parallel loads at 
right angles to their axes, the loads supposed to be acting 
in a plane which contains the axis of the beam in each case. 
Under such conditions there will clearly be no direct uni- 
form compression on any normal section of a beam. This 
is equivalent to assuming that the flexure is produced by an 
indefinitely small force acting parallel to the axis (or at 
right angles to a normal section) of the beam and at an 
infinite distance from the latter. 

It is clear, since the product of the distance of the point 
of application of a force normal to the cross-section from 
the centre of gravity of the latter multiplied by the dis- 
tance of the neutral axis from the same point, but on the 
opposite side from the point of application of the loading, 
must be equal to the square of half the diameter of the 
ellipse of inertia, that if that square be divided by 



736 MISCELLANEOUS SUBJECTS, [Ch. XVI. 

infinity, the distance of the point of appHcation of the 
load from the centre, the quotient will be zero, i.e., the 
neutral axis must pass through the centre of gravity of 
the section. 

This condition is further equivalent to taking any finite 
loading at right angles to the axis of the beam, as in the 
ordinary cases of engineering practice. The stresses found 
in the normal section in such cases will be the direct tension 
and compression with intensity varying directly as the 
normal distance from the neutral axis with the accompany- 
ing shears, as in the common theory of flexure. 

The preceding investigations show, however, that with 
unsymmetrical sections the neutral axis, while passing 
through the centre of gravity of the section, is not at right 
angles to the plane of loading, unless that plane happens 
to contain one of the two principal axes of inertia of the 
section. 

Let the Z-bar section shown in Fig. 2 of the preceding 
article be considered and suppose that the loading acts in 
the vertical plane ZZ' , the latter line passing through the 
centre of gravity C of the cross-section. It may be con- 
sidered that the Z-bar is supported at each and on the lower 
surface HL of the lower flange. Inasmuch as the bending 
moment acts in the plane ZCZ' the neutral axis will be 
drawn through the centre C parallel to the tangents to the 
ellipse "where the line ZZ' cuts the latter, as shown at g 
and at the opposite end, not lettered, of the vertical diameter. 
The diameter A'B' is then the neutral axis desired. The 
line Ch drawn at right angles to ZZ' may be considered 
the axis of the external bending moment to which the beam 
is subjected. The angle between Ch and the neutral 
axis is a, as shown. 

If the coordinate x be taken as at right angles to the 
neutral axis A'B' , and if dA represent an element of the 



Art. 1 1 6.] GENERAL FLEXURE TREATED BY CORE METHOD. 737 

normal section of the beam, then the distance of that element 
from the neutral axis measured parallel to ZZ' will be 
X sec a. If k is the maximum intensity of stress at any 
point of the section, that stress will occur at L or T, where 
the value of x=n is the greatest for the entire section. 
The distance of that point parallel to ZZ' will be n sec a. 
If M is the value of the external bending moment acting 
in the plane ZZ' , dM may be written : 

dM = xsec a-dA'Xsec a. . . . (i) 

n sec a . 

If / is the moment of inertia of the section about the 
neutral axis, 

M = CdM =-I sec a- =-/A sec a. . . (2) 
J n n 

In Fig. 2 the line ZZ' cuts at d the side DF of the core. 
Let the distance dC be represented by /. Then, as shown 
in the preceding article, 

jCZ'^Cg. 

But the radius of gyration of the section about the axis 
A'B' has been shown in Art. 81 to be equal to the normal 
distance r' between the neutral axis and the parallel tangent 
to the ellipse drawn at g. 

Cg = / sec a. 

■ It has already been seen that CZ' is equal to n sec a. 

r'^ sec a = nj. 

If this value of r''^ sec a be substituted in the third 
member of Eq. (2) there will result, 

M=kAj (3) 



738 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

Eq. (3) is the expression for the external bending moment 
in terms of the greatest intensity of stress in the section, 
the area of that section, and the distance j from the centre 
of the section to the side of the core as constructed by the 
methods explained in the preceding section. Although the 
construction has been made with the Z section the method 
of procedure is precisely the same for any form of section 
whatever. 

Component Moments. 

By again referring to Fig. (2) of the preceding article 
it will be seen that M cos a is that component of the external 
moment whose axis is parallel to the neutral axis, while 
the component M sin a has an axis be at right angles to the 
neutral axis, but lying in the plane of the normal section 
of the beam. The former component produces the bending 
stresses about the neutral axis, the maximum intensity of 
which is k and a deflection normal to it; the latter compo- 
nent moment tends to produce an oblique movement of 
the beam in consequence of its unsymmetrical section.. 

This tendency in oblique flexure, especially marked with 
unsymmetrical sections, is always toward that position 
in which the least radius of gyration of the section (repre- 
sented by the least semi-axis of the ellipse of inertia) is 
found in the plane of bending, i.e., that plane in which the 
bending moment acts. , 

In Fig. 2 of the preceding article ZZ' is the plane in which 
the vertical loading acts, and it is clear that the plane in 
which the resultant bending compression on one side of the 
neutral axis A'B^ and the resultant bending tension on the 
other side act is not the plane in which ZZ' lies, but inclined 
somewhat to the right of CZ. Inasmuch as these two 
planes are neither the same nor parallel, there must be 
combined with the couple producing pure flexure such a 



Art. 117.J PLANES OF RESISTANCE IN OBLIQUE FLEXURE. 739 

couple as to make the resultant external moment equal 
and opposite to the internal resisting moment, and the 
component of M represented by be is such a couple, Ce 
representing the couple producing pure flexure about 

These analytic considerations show how essential it is 
to give careful consideration to the principles governing 
oblique or general flexure for loads not in a plane of 
symmetry of a beam and for unsymmetrical sections. 

The method of finding the location of the plane of 
resistance of the bending stress existing in any normal 
section of the beam will be given in the next article. 

Art. 117. — Planes of Resistance in Oblique or General Flexure. 

The preceding treatment of general flexure has shown 
that the plane of action of the external bending moment 
will not in general coincide with the plane in which the 
internal resisting couple acts. The plane of the external 
bending moment is supposed to pass through the axis of 
the beam assumed to be straight. If this external bend- 
ing couple is to produce pure flexure it must be in equilib- 
rium with the internal moment produced by the stresses 
in any normal section, and that requires that the two planes 
of action shall either coincide or be parallel. 

Let it be supposed that the 6X3jXf-inch steel angle 
section shown in Fig. i represent any unsymmetrical sec- 
tion, and let it also be supposed that G^F is the neutral 
axis of the section, G being the centre of gravity; then let 
GX and GY be the axes of rectangular coordinates negative 
when measured to the left and downwards. The stresses 
above GY will be supposed compressive, and those below, 
tensile. The intensities will be assimied to vary directly 
as the normal distances from GY as in the ordinary theory 



740 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

of flexure. The centre of all the compressive stresses will 
be taken at C and at T for the tensile stresses. The plane 
whose trace is CT will be called the plane of resistance, 
while AB^ will be taken as the plane of action of the 
external bending moment. In other words, if the angle 
were to carry vertical loading as a beam AB should be 
vertical with the lines of cross-section correspondingly 
inclined. 

If xi and a'l are the coordinates of the centre C of the 
compressive stresses in the section and if a is the intensity 
of stress at a unit's distance from the neutral axis GY, 
eqs. (i) and (2) will immediately result: 

r Cyaxdxdy j Cxydxdy j^ 
C ( axdxdy C fxdxdy Qi 

( j xaxdxdy \ \ x~dxdy j'^ 
\\ axdxdy ( j xdxdy Qi 

The quantities /i and I'l sue the so-called ''product 
of inertia " and the moment of inertia of that part of the 
cross-section lying above GY, while Qi is obviously the 
statical moment of the same part of the cross-section in 
reference to the same axis. 

If the subscript 2 be used for the corresponding quanti- 
ties relating to that part of the section below GY, eqs. (3) 
and (4) will at once result, the negative sign being used 
in the second member because the coordinates are negative : 

^2=-^, (3) 

^2=-^^ (4) 



Art. 117.] PLANES OF RESISTANCE IN OBLIQUE FLEXURE. 741 

Q, I and / represent quantities belonging to the whole 
cross-section, then, since G is the centre of gravity of that 
section, 

Qi=Q2=Q\ 

I\+I'2=I\ 
Jl+j2=J. 




Fig. I. 



It is desired to find the straight line joining C and T, 
and in order to do that the general equation of a straight 
line may be written as follows: 



x+by—c. 



(s) 



742 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

If yi and Xi taken from eqs. (i) and (2) be first written 
in eq. (5) and then >'2 and X2 from eqs. (3) and (4), and if 
the second of the equations so formed be subtracted from 

the first, there will result: b = — j. 
Then eq. (5) will take the form 

x=jy-\-c (6) 

In Fig. I suppose a line parallel to CT drawn from G 
to B. If the ordinate xi be produced upward, the line 
BC = Gc' will be determined. If in eq. {6) y =0, x=c = Gc' = 
BC. The triangles with the bases yi and >'2 will then be 
similar and that similarity will be expressed by the following 
equation, remembering that x and y are negative: 

X\—C —X2 + C f . 

— ^= — - (7) 

Substituting the values of x\, yi, X2 and y2 established 
above there will result the following value of c: 

JQ • 

Placing this value of c in eq. (6), 

"^'f JQ ' .^^^ 

This is the equation of the line CT, Fig. i, drawn through 
the centres of the tensile and compressive resisting stresses 
acting in the normal section, i.e., it is the trace of the plane 
in which the resisting couple acts. The tangent of the 



Art. 117.] PLANES OF RESISTANCE IN OBLIQUE FLEXURE. 743 

dx I 
angle which it makes with the neutral axis GY is -r- =-. 

dy J 

If GY is one of the principal axes of inertia of the section 

dx 
J =0 and -:- becomes infiniteiy great, i.e., in that case 

the line GT is at right angles to 6^y and it will presently 
be shown that it will pass through G, the centre of gravity 
of the section. 

If :v=o in eq. (8), 

^0= -^ ^ =Gc (9) 

The distance Gc' is on the negative side of G. Again if 
x=o, there will result : 

y= _jQ =G^ (10) 

These coordinates Ge and Gc' shown in Fig. i give two 
points e and c' in the desired line CT, which must agree 
obviously with the points C and T as found by computa- 
tions. 

If Ge should be zero, eq. (11) will result: 

ja'2-I\j2=o (11) 

Inasmuch as the moments of inertia I\ and I' 2 will 
always have real values for an actual section, in general 
if eq. (11) holds true, then must Ji=j2=o. That condi- 
tion will of course exist for the principal axes and for the 
case where at least one of the coordinate axes is an axis 
of symmetry of the section. 

Although the figure used for the establishment of the 
preceding formulae is the normal section of a steel angle, 
those formulae are completely general and are applicable 



744 MISCELLANEOUS SUBJECTS. [Ch. XVL 

to any form of cross-section whatever, as indicated by 
eqs. (i) and (2) and all the equations following. 

It is thus seen that if the plane of action AB oi any 
external loading producing flexure of a beam with unsym- 
metrical cross-section is parallel to the plane whose trace 
is CT, there will be pure bending only as the external bend- 
ing moment has the same axis as the couple formed by the 
internal stresses. The planes of the external bending 
moment and that of the internal resisting stresses may in 
some cases coincide. 

If the steel angle shown in Fig. i is to act as a beam 
under vertical loading in pure flexure, the end supports 
should be so formed as to make the lines AB and CT verti- 
cal. In general, whatever may be the cross-section of a 
beam, the latter should be so held at its points of support 
that the loading will produce pure flexure. If the section 
of the beam has an axis of symmetry, the plane of loading 
may be taken through the axes of symmetry of the cross- 
section. 

Example. The application of the preceding formulae 
may be illustrated by using the 6 X si-inch, 22.4-lb. steel 
angle shown in Fig. i. The thickness of each leg is .75 
inch. By using eqs. (i) and (2) there will at once result: 

7'i=9.4i; r2 = i3.94; Ji=5-47\ 
72=3-04; 7=8.51; = 5-484 

Inserting these values in eqs. (i), (2), (3) and (4) there 
will result: 

yi=i in.; ^^1=1.72 ins.; j2 = -.555 in.; 

:;t:2 = —2.54 ins. ; :ro =— 1.02 ins. ; 3/0 = .372 in. 

These coordinates are laid off in Fig. i, as shown, so as 
to locate the four points C, e, c' and T. In making these 



Art. ii8.] DEFLECTION IN OBLIQUE FLEXURE. 745 

computations it should be remembered that I'l and I' 2 
are moments of inertia of areas, one of whose sides coin- 
cides with the axis of y and that the same observation is 
also true of the quantities, /i and J2, as well as Q. 

Art. 118. — ^Deflection in Oblique Flexure. 

The general case of deflection of a beam with unsym- 
metrical cross-section, or of a beam with symmetrical 
cross-section but loaded obliquely, may readily be found 
by the aid of the ordinary formulas for flexure used in 
connection with the preceding investigations. The requisite 
treatment may be well illustrated by considering the case 
of a 6 X3i Xf-inch steel angle, the section of which is shown 
in Fig. I to be same as that used in the preceding article. 
Such an angle may be considered to be used as a beam 
in roof work or for some other similar purpose with the 
6 -inch leg placed in a vertical position. It will be assumed 
that the span length is 15 feet = 180 inches and that the 
angle is to carry as a beam a uniform load of 200 pounds 
per linear foot. The data given in an ordinary handbook 
on steel sections will show the position of the centre cf 
gravity G of the section and enable the ellipse of inertia to 
be constructed as in Fig. i. 

The maximum radius of gyration represented by the 
greater semi-axis of the ellipse is 1.97 inches, while the least 
radius of gyration at right angles to the preceding and 
represented by the smaller axis is .75 inch. The load 
acts in a vertical plane passing through the axis G. The 
various dimensions of the cross-section required in the 
computations are" all shown in Fig. i. 

By drawing vertical tangents on opposite sides of the 
ellipse, the neutral axis A'B' drawn through the points of 
tangency and the centre G of the ellipse is determined. 



746 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

This neutral axis of the section makes the angle, 46° 30', 
as carefully measured on the diagram, with the horizontal 
axis of Y. By drawing a tangent to the ellipse parallel 
to A'B^ the radius of gyration about the neutral axis is 
found to be i.i inches, i.e., the normal distance between, 
the neutral axis and the parallel tangent to the ellipse. 

The greatest deflection of the angle beam will be found 
at the centre of span at which the moment of the external 
forces is 

^^ 200X225 ^^^ =67,500 in.-lbs. . . . (i) 
8 

The component moment, as shown in the preceding 
article, with axis parallel to the neutral surface, is 

M cos a = .6884^=46,467 in.-lbs. ... (2) 

The component moment having an axis at righ^" angles 
to the neutral axis is, similarly, 

M sin q: = . 7 2 5471^ = 48,964 in. -lbs. ... (3) 

The actual flexure is produced by the first of these com- 
ponents M cos a. The deflection produced by it will obvi- 
ously be normal to the neutral axis, and it can be computed 
by the ordinary formula for the deflection at the centre 
of span of a beam simply supported at each end and loaded 
uniformly throughout its length, the uniform load to be 
taken in this case as 200 cos a = 138 pounds per Hnear foot. 
If g is the load per linear foot of span, the usual expres- 

sion for the centre deflection is w= ^ -r^-r - Substituting 

3S4EI 

138X15 for g/ in the form.ula, / = i8o inches, £=30,000,000, 



Art. ii8.j 



DEFLECTION IN OBLIQUE FLEXURE. 



747 



and 7 = 7.94 (moment of inertia of section about the neutral 
axis) there will result: 



w =0.66 inch. 



(4) 



As- cos a = .6884 and sin a = .7254, the vertical deflec- 
tion =.66 X. 6884 =.454 inch; and the horizontal deflec- 
tion = .66 X.7254 = .479 inch. 




Fig. 



It is thus seen that the horizontal deflection slightly 
exceeds the vertical, in consequence of the major axis of 
the ellipse of inertia being slightly inclined to a vertical 



748 MISCELLANEOUS SUBJECTS [Ch. XVI. 

line, thus causing the inchnation of the neutral axis of 
the section to be relatively large. 

Precisely the same general treatment would be followed 
for any form of cross-section or any other amount or dis- 
position of loading. 

In the preceding article where the same angle was so 
held as to make the plane of loading parallel to . that of 
the resisting couple, the horizontal diameter of the ellipse 
drawn through G is the neutral axis corresponding to the 
conjugate diameter DF, parallel to the trace of the plane 
of the resisting internal couple as determined in that 
article. The normal distance, 1.95 inches, between the hori- 
zontal diameter through G and the horizontal tangent at 
F is the radius of gyration corresponding to the horizontal 
neutral axis through G. As the area of cross-section of 
the steel angle is 6.56 square inches, the moment of inertia 
corresponding to the horizontal neutral axis through 
G is J =6.56 X 1.95" =24.93, "the moment of inertia of the 
cross-section about the neutral axis A'B\ Fig. i, is 
7 = 6.56X1.1^ = 7.94. The distance from the horizontal 
neutral axis through G to the extreme fibre is 3.82 inches, 
while the corresponding distance of the extreme fibre from 
A^B'' is 2.3 inches. Hence, the resisting moment for the 
horizontal neutral axis through G is 

3.82 
. For the neutral axis A'B^: 



2.3 
Hence —p = i.g. In other words, the same angle placed 



Art. 119.] ELASTIC ACTION UNDER DIRECT LOADING. 749 

SO as to take the vertical loading in a plane parallel to the 
resisting internal couple will offer nearly twice as much 
bending resistance with the same extreme fibre stress as 
when placed with the longer leg vertical. Economic use 
of the metal as well as avoidance of unnecessary deflection, 
therefore, requires that the beam of unsymmetrical section 
shall be so held at its supports as to make the plane of 
loading parallel to the resisting plane and as nearly parallel 
to the greater axis of the ellipse of inertia of the cross- 
section as possible. 

Art. 119. — Elastic Action under Direct Loading of a Composite 
Piece of Material. 

Let it be supposed that a combined straight or cylin- 
drical piece of material with length L is subjected to the 
direct stress of either tension or compression. If the total 
area of cross-section is A, it may be assumed to be composed 
of the following parts : 

A I =area of cross-section with modulus of elasticity Ei; 

A 2 =area of cross-section with modulus of elasticity E2', 

A3 =area of cross-section with modulus of elasticity E3; 
etc., etc. 

Then will 

A=Ai-\-A2-\-A3-\-etc (i) 

Let the total load P act parallel to L and let / be the 
strain per unit of length of the piece, i.e., the unit strain, 
then will IL be the total lengthening or shortening of the 
piece. Under these conditions every part of the piece 
will be subjected to the same rate of longitudinal strain 
and the following equation may be at once written: 



7SO MISCELLANEOUS SUBJECTS. [Ch. XVI. 

EilAi-\-E2lA2-\-E3lA3-]-etc.=P=ElA. . . (2) 

Hence, 

1 = CO 

Also the first and third members of eq. (2) will give 

eq. (4) : 

^ £iAi+£:2^2+£3^3+etc. , . 

E=- J . . . (4) 

Eq- (3) will give the lengthening or shortening of each 
unit of length of the piece under any assigned load P, the 
moduli of elasticity of the areas of the different parts of 
the section being known. 

The modulus of elasticity E given by eq. (4) may be 
considered a mean or average modulus or an equivalent 
value for the actual moduli, as the same longitudinal strain 
would be yielded by a piece of uniform material having 
that modulus of elasticity and the same area of cross- 
section as the composite piece. 

Art. 120. — ^Helical Spiral Springs. 

A spiral spring like that shown in Fig. i takes its load 
at the ends as indicated at A and B. In the general case 
there may be applied at each end a single load P and a 
couple, or either a force or a couple alone may act. The 
analysis will be so written as to include concurrent force 
and couple or either one separately. The following nota- 
tion will be employed: 

R = radius of spiral, Fig. i ; 

(j) = pitch angle of spiral, Fig. i ; 

z = axial elongation or compression of spring under load- 
ing; 



Art. 120.] HELICAL SPIRAL SPRINGS. 751 

/ = length of spiral ; 

r = radius of spiral wire ; 
P = axial load, Fig. i ; 

M = moment of applied twisting couple or torque, as- 
sumed to be a right-hand moment ; 

u — unit strain at unit distance from, the neutral axis 
in bending or flexure ; 

a = angle of torsion (unit strain at unit distance from 
axis of piece in torsion) ; 

T = total twist or rotation of spring measured on central 

cylinder of spiral ; 

T 
T =—= angle of twist of spring in radians. 

The force P will be considered positive when it stretches 
the spring as shown in Fig. i. If the force P compresses 
the spring it must have the negative sign in all the following 
analysis. 

The moment M will be considered a right-hand moment 
when it twists the spiral so as to bring the helical parts 
near together, i.e., tightens the spiral. It should be re- 
membered that all parts of the spiral are uniformly stressed 
or bent. The cross-section of the spiral rod will be con- 
sidered circular, although the general analysis is adapted 
to any form of cross-section. 

The load P produces a moment Mi about the centre of 
any section of the spiral rod given by 

Mi=PR (i) 

The axis of this moment is a horizontal line through 
the centre of the section and tangent to the central cylinder 
of the spiral shown by a broken-line circle in the lower 
part of Fig. i. li A, Fig. 2, be the centre of the section 



752 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



considered, KL may be taken as the axis of the moment 
PR. If AK, therefore, represent by a convenient scale, the 
moment Mi=PR, AG and GK (drawn perpendicular to 
AG) will represent by the same scale the component mo- 
ments of Ml about those lines as axes passing through 
the centre of the section. As the axis ^G* is the axis of 




Fig. I. 



Fig. 2. 



the -spiral rod, it represents a torsion moment. Similarly 
GK represents a bending moment as it lies in the section 
and, in fact, is a neutral axis. Hence, if the subscripts 
t and h mean torsion and bending, 



And, 



AG=M\=Mi cos (f>. 
GK=M\=-M sin ct>. 



(2) 
(3) 



Art. I20.] HELICAL SPIRAL SPRINGS. 753 

The moment — M sin has a negative sign because the 
triangle AKG, Fig. 2, shows that it will tend to untwist 
the spiral of Fig. i, which is opposite to a positive effect. 

The right-hand moment M will act at the centre of 
section of the spiral rod about an axis parallel to AC, 
Fig. I, i.e., about BD, Fig. 2, and AB may represent that 
moment. Its two components will be: 

BF = M'\=M sin cf>, .... (4) 

AF=M",=M cos d, (5) 

The resultant moments of torsion and bending at the 
section considered will therefore be: 

Mt=Mi cos <t>+M sin (j>, ... (6) 

M^=M cos 0-Afi sin 0. ... (7) 

By the common theory of torsion (correct for a cir- 
cular section only) if G is the modulus of shearing elasticity, 
the angle of torsion, or unit strain a, is 

moment Mi cos (/)+-M'sin 4) ,„s 

«= -j-= ■ y. . ... (8) 



Evidently, Q=G — (for circle); and Q=G— (for 
2 6 

square) . 

If the exact theory of torsion is used for other sections 
of the spiral rod than circular, the corresponding value of 
a must be introduced, but no other change is needed. 

In the same manner, if E is the modulus of elasticity 
for direct stress, I the moment of inertia of the section 



754 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



Hadl cos 



about its neutral axis, and if Q' =EI =E^^ (for circular 

4 

section) or Q' =E — (for square section), the unit strain, 
12 

Uy for bending is, 

moment M cos 0— Mi sin , . 

«=^^= Q, .. . . (9) 

The quantities a and u are unit motions giving to the 

spiral spring corresponding motions of rotation and axial 

lengthenings or shortenings. 

The torsion moment Mt will cause one end of an indefi- 
nitely short length dl of the 
Radi sin4> spiral rod to rotate through 
the angle adl, inducing a 
movement of that end, rela- 
tive to the axis of the spiral, 
perpendicular to the axis of 
the rod, equal to Radl, as 
shown by Fig. 3. The hori- 
zontal component of this 
-Budi cos movement tangent to the- 
spiral cylinder is, Radl sin 4>, 
or for each unit of length of 
the rod, Ra sin 0. As the 
state of stress is uniform 
throughout the spiral rod, the 

total circumferential twist of the spiral spring due to 

torsion is 




Fig. 3. 



•Jtotdi sin ' 



-Rudl 



Fig. 4- 



r=Rla sin <t>=Rt 
And the angle of twist is 



Ml cos (/)+Msin 



Q 



sm (/>. 



(10) 



r=^=/ Q sm<#.. 



(loa) 



Art. 120.] HELICAL SPIRAL SPRINGS. 755 

The axial component of the same movement, as shown 
by Fig. 3 is, Radl cos </>. Hence the total axial movement 
due to torsion is 

, „jMi cos </)+M sin <A x 

z'=Rl — ^ -^cos0. . . (^11) 

The movement of a normal section of the spiral rod, 
relative to the axis of the spring, due to bending about its 
neutral axis parallel to GK and AF, Fig. 2, is illustrated by 
Fig. 4. That movement will be parallel to the axis of the 
rod and the broken-line triangle showing it and its com- 
ponents is moved vertically to clear it from the centre line 
of the rod. The horizontal component representing the 
tangential or rotating movement due to bending is seen to 
be 

Rudl cos cf). 

Or, for the entire length / of the spring, 

^„ „,ikf cos (^— Ml sin , , v 

T" =Rl ^ cos 4). ... (12) 

The angular twist is 



;/ 



T" M cos -Ml sin ^ , ^ 

=— =/ —, COS0. . . (12a) 



Similarly, the axial component of the movement due 
to bending, as shown in Fig. 4, is 

— Rudl sin 4). 

This value is negative, as the axial motion is downward 

and opposite to that due to torsion shown in Fig. 3. 

Hence, 

,, ^,M cos 0— Ml sin (^ . ^ , >. 

2 = -Rl ^, sm 4>. . . (13) 



756 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

The angle of twist of the spring under loading will be 
the sum of the second members of eqs. (loa) and (12a): 

71/r 7 • , , /i I \ , Ti/Tz/sin^ , cos^ (t>\ f N 

T=Mi/sm0cos0(^^-^j+M/(^-^+--Q7^j. (14) 

The circumferential motion of the spring will be 

T-Rr (15) 

The axial extension or compression of the spring will 
be found by the aid of eqs. (11) and (13) : 

+Msin0cosc/)(^-^M. (16) 

Eqs. (6) and (7) will enable any spiral spring to be 
designed to perform a given duty such as to carry a pre- 
scribed load or serve the purposes of a dynamometer, while 
eqs. (14) and (16) will give the distortions of the spring, 
either angular or axial. 

If 5 is the greatest intensity of torsive shear in a normal 
section of the spiral rod at the distance r from the centre, 
while /p is the polar moment of inertia of the section, 

. M,='-U. ...... (17) 

r 



irr 



For a circular section, 1^ = '^^. 

2 



54 
For a square section, Ip =—(£» = side of square). 

6 

Eq. (17) gives: 

s=^. ■ (x8) 



Art. I20.] HELICAL SPIRAL SPRINGS. 757 

When 5 is given, 

r=\ ^ (circular section). , . . (i8a) 

\ tS 

In both eqs. (6) and (7), Mi and M are known quanti- 
ties, as they are the given loads. 

Again if k is the intensity of stress in the most remote 
fibre at the distance di from the neutral axis, and if I is 
the moment of inertia of the section about the neutral axis, 

^=—J- (^9) 

When k is given, 

c^i =f =x/^-* (circular section). . . (loa) 

The two intensities 5 and k exist at the same point, 
and they are to be used to determine the greatest intensities 
of stress in the cross-section of the spiral rod precisely as was 
done in Art. 10. 

By eq. (2) of that article, the greatest and least inten- 
sities of stress (principal stresses of opposite kinds) will be : 



k I k^ 
max. intensity =-+a| 52 H — (tension) 



2^4 



min. intensity = — \/^^H — (compression). 
2 ' 4 

At the opposite end of that diameter of section of the 
rod normal to the neutral axis where k is compression, 
the above " max. intensity " will be compression also, 
and the " min. intensity " will be tension. 



758 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

The planes on which these principal stresses act are 
given by eq. (3) of Art. (10) : 

, 25 

tan 2a = — —. 
k 

The greatest shear at the same point is given by eq. 
(6) of Art. (9) ; i.e., its intensity is half the difference of the 
principal intensities, or, 

max. — min. 

P,= — - — . 

There are a number of special cases which may easily 
be developed from the preceding general analysis. 



Small Pitch Angle. 

' If the pitch angle </> is so small that sin <}> may be con- 
sidered zero without essential error, 

sin 0=0 and cos = i. 

Eqs. (6) and (7) then give: 

Mt=Mi=PR\ ..... (20) 

M6=M . (21) 

From eqs. (14) and (16): 



PRH ( X 



Art. 120.] HELICAL SPIRAL SPRINGS. 759 

Rotation of Spring Prevented. 

In this case twisting of the spring is prevented, or r =0. 
Eq. (14) then gives: 

T\/r M sin (jy cos <I>(Q' -Q) - .. 

^=-^^Q'sin^0+Qcos^0- • • • ^^4) 

Substituting this value of M in eq. (16) : 

^_7^P2/f cQS^ ^ I si^^ ^ (sin cos cf>Y{Q' -QY \ /a 
'"1 Q ^ Q' (0^sin^0+Qcos^c/>)QQt ^ ^^ 

The torsion moment Mt, eq. (6), and bending moment 
Mb, eq. (7), are to be computed by using the value of M 
given in eq. (24). 

Axial Extension or Compression Prevented. 
By making 2 =0 in eq. (16), 

■ M.= -Mj^^^|fcg. . . . (.6) 

Qcos2 0+Qsm2(/) 

The angle of twist then becomes: 

sin2 , cos2 4> (sin 4> cos0)2(Q' -Q)2 



" ^\ Q ^ Q' (Q^cos2c/>4-Qsin2 0)Q^Qr ^'^^ 

For circular or square sections Q'— Q = ( 6^)(— or—) 

and the square of the latter alternative factor is common 
to {Q' —QY and Q'Q in the second number of eq. (27), thus 
canceling and simplifying the numerical application of 
that equation. 

In computing Mt and Mb, eqs. (6) and (7), the value 
of Ml given by eq. (26) is to be used. 



76o MISCELLANEOUS SUBJECTS. [Ch. XVI. 

This form of helical spring is employed in the transmis- 
sion dynamometer. 



Work Performed in Distorting the Spring. 

The work performed in producing the angular and axial 
distortions r and z by the moment M and force P is easily 
found by the aid of eqs. (14) and (16) or corresponding 
equations for special cases. The couple whose moment 
is M performs work in twisting the spring through the arc 
T (measured at unit distance from the axis of the helix) 
expressed by 

, w,=Mi. ...... (28) 



The force P performs work in extending or compressing 
the spring the distance z given by the equation 



W^=^ . (29) 

2 



The total work done in the general case will then be : 

W = WtVW^=^{Mr+Pz). . . . (30) 

For special cases, as already indicated, the corre- 
sponding value of T and z must be used in eq. (30). 

In wTiting the preceding equations it has been assumed 
that both M and P are gradually applied. If they were 
suddenly applied, the distortions would be 2t and 2Z and 
oscillations having those amplitudes would be set up. 
The periods of the amplitudes would depend upon the' 
masses moved. • 



Art. 121.] PLANE SPIRAL SPRINGS. 761 

Art. 121. — ^Plane Spiral Springs. 

A plane spiral spring may be represented by Fig. i. 
The outer end is fastened at B, but the inner end is secured 
to a rotating post or small shaft at C. The spring or coil 
is " wound up " to an increasing number of turns by apply- 
ing a couple to the shaft C, as in winding a clock. 

As a couple only is applied at C, every section of the 
spring is subjected to bending by the same couple, i.e., 
there is a uniform bending moment throughout the entire 
spring. This uniform condition of stress makes the analysis 
of this spring exceedingly simple if the thickness of the metal 
is small. , As the spring is a spiral beam subjected to uni- 
form bending, the analysis, to be perfectly correct, should 
be based on that for curved beams. That procedure would 
introduce much complexit}^, and as the thickness of the strip 
of metal constituting the spring is small compared with its 
radius, no essential error is committed in neglecting the 
effects of curvature. The usual cross-section of this type 
of spring is a much elongated or narrow rectangle, the 
greater dimension of the rectangle being parallel to the 
axis of the couple or perpendicular 
to the plane of the spring. 

If u is the unit strain at unit 
distance from the neutral axis of a 
section of the spring, I the moment 
of inertia of the same section about 
the neutral axis, and E the modulus 
of elasticity, while M is the moment 
applied at C, Fig. i, W//////////////M 

M=EJw= constant. . . (i) 




Fig. 



If / is the total length of the spring and /S the total 
angular distortion for that length, then will udl be the 



762 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

change of direction or angular distortion for each element 
dl. Hence, 

Mdl=EIudl=EIdl3. ...... (2) 

Integrating : 

Ml^EI^; and ^^g. . . . (3) 

With the thin metal used / is small and 13 may be a 
number of complete circles, perhaps sufficient to wrap the 
spring closely around the shaft C. 

If the moment M is applied gradually, the work done in 
producing the total angular distortion /3 is 

This is the same as the expression for the work performed 
in bending a beam by a moment uniform throughout its 
length. In fact the plane spiral is simply a special case of 
flexure, the bending moment being uniform. 

If the moment M should be applied suddenly, the total 
angular distortion would be 2/3, and oscillations having 
that amplitude might be set up. 

Art. 122. — ^Problems. 

Problem i . — A helical spring having a diameter of helix 
of .3 inches and composed of twelve complete turns of a 
f-inch round steel rod sustains an axial load of 45 pounds. 
Find the axial deflection of the spring and the greatest 
intensities of torsive shear and bending tension and com- 
pression in the rod. 

P = 45 lbs. ; ^ = I • 5 ins. ; = 15°; 1 = 117 ins. ; 

£^ = 30,000,000; (7 = 12,000,000; • r=i^in. 



Art. 122. 



PROBLEMS FOR ARTS. 120 AND 121. 



763 



Mi=PR=6S,s in.-lbs.; 



,7rr 



Q=G — = 23,373; 
2 



M=o] 

Q =E — = 29,217. 
4 



Substituting these quantities in eq. (16): 

.=3Xii7(^^^+^^^^)68.s=..746m. 

\23, 373 29,217/ 

By eqs. (6) and (7) : 

Mt=Mx cos 0=66.2 in.-lbs.; 



and 



M6= -Ml sin 0= —17.74 in.-lbs. 



Trr 



TTf^ 



Since 1^ = '-^-^ and 7= — , eqs. (18) and (19) give: 
2 4 

5 =9460 lbs. per sq.in. torsive shear; 

k =3432 lbs. per sq.in. greatest bending stress. 

Problem 2. — Design a helical spring for a transmission 
dynamometer for 8 H.P., at 90 revolutions per minute. 
Axial distortion of the spring is prevented, or z=o. Let 
low working stresses and other data be taken as follows* 



Iz = 16,000 lbs. per sq.in. 
i? = 3 ins. ; = 11° 

G = 12,000,000; 
MX9oX27r=8X33.ooo 

Q=G— and 
2 



5 = 12,000 lbs. per sq.in. ; 
.*. sin (/) = .i9i and COS0 = .982; 

E =30,000,000. 
M=466.8 ft.-lbs. =5602 in.-lbs. 



Q'=£^. 



Eq. (26) then gives: 

Ml = — 212 in.-lbs. 



764 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

By eqs. (6) and (7) : 

M/ = 862 in. -lbs. ; and Ms = 5541 in. -lbs. 

Solving eqs. (18) and (19) for the radius of the rod: 
By eq. (i8a), r = .s6 in.; and by eq. (19a), r = .'j6 in. 
Bending of the rod, therefore, requires the- greater 
radius, and r = .'/6 in. will be taken. 

Eq. (17) gives the greatest torsive shear in a section: 

5 = 1250 lbs. per sq.in. 

The equations following eq. (19) now give: 

max. intensity = -f 16,097 lbs. per sq.in.; 
min. intensity = —97 lbs. per sq.in. 

The spring will be assumed to have twelve complete 
turns, so that its length will be: 

/ = 27r3 X 12 Xsec = 230.5 ins. 

The twist r at unit distance from the axis of the helix 
now becomes : 

r = .i59in. 

At the distance of 10 inches from the axis the twist 
would be 1.59 inches, but the spring is too stiff to be very 
sensitive. A higher working stress k may properly be taken. 

If in the same problem there be taken 120 revolutions 
per minute and an alloy steel for which the working stresses 
/^ =40,000 pounds per square inch and 5=30,000 pounds 
per square inch may be prescribed, then by using the 
results already established : 

•M=-^X 5602 =4200 in. -lbs.; 
120 



Art. 122.] PROBLEMS FOR ARTS. 120 AND 121. 765 

Mi = -JX2i2 = — 159 in.-lbs. ; 
Mr = f X862 =647 in.-lbs.; 
-^6 = 4X5541 =4156 in.-lbs. 

For shear: r =^/-X— X.36 =.67 X.36 =.24 in. 

\4 2.5 

For bending: r =a/-X— X. 76 =.67 X. 76 =.51 in. 

y A 2.^ 



4 2.5 
159 



(.67)^ 



795. 



At the distance of 10 inches from the axis of the helix 
the twist would be ioX.795=7.95 inches. 

Problem 3. — What will be the angular distortion j8 
of a plane spiral spring i inch by -^-^ inch in section and 
20 inches long if the distorting moment is 10 inch-pounds. 
Eq. (3) of Art. 121 gives: 

10X20 10 X20 X12 Xi2i;,ooo 

^=- — =. ^ =10 

30,000,000 Xi 30,000,000 

(about 1 1 complete turns). 
The fibre stress is 

10 X-- 
k = = 1 50,000 lbs. per sq.m. 



12 X 125,000 



Art. 123.— Flat Plates. 

The correct analysis of stresses in loaded fiat plates even 
of the simplest form of outline has not yet been made suf- 
ficiently workable for ordinary engineering purposes, either 
for plates simply resting on edge supports or with edges of 
plates rigidly fixed to their supports. It is necessary, 



766 MISCELLANEOUS SUBJECTS. fCh. XVI. 

therefore, to combine simple, but approximate analysis 
based on reasonable assumptions, with experimental results 
so as to obtain workable formulae. The following pro- 
cedures, due chiefly to Bach and Grashof, are commonly 
employed in treating flat plates: 

Square Plates — Uniform Load. 

In Fig. I let A BCD represent a square plate simply 
resting on the edges of a square opening. Tests of such 
y plates by Bach have shown that when 
increasingly loaded they will ulti- 
mately fail along a diagonal, as AB. 
Let the plate be uniformly loaded 
with p pounds per square unit, then 
let moments be taken about the diag- 
onal AB. If & is the side of the 



Pjg. I. square, the load on the triangular 

half of the square is - — , and the distance of its centre 

2 

from AB is ^^ sin 45° =.23 6^. The upward supporting 
forces or reaction on the sides AD and DB will also be 

half the load on the plate, — , and its centre will be at the 

2 

distance ^=.3546 from AB. Hence the moment 

2 

about AB will be: 

M=^(.SS4b-.236b)=.oS9pb^. . . . (i) 

If h be the thickness of the plate, the moment of inertia 
I about its neutral axis will be: 

J b sec 45° h^ ^..^ , X 

/= !t^ =.iiSbh^ (2/ 

12 



Art. 123.I SQUARE PLATES. 767 

The ordinary flexure formula then gives for the greatest 
intensity of bending stress k, assuming it to be uniform 
throughout the diagonal section, 

^^~T^IW V ^^^ 

Or, if the thickness is desired, 

k^h^^. ....... (4) 

Eq. (4) gives the thickness of plate required to carry 
the unit load p when the working stress is fe. 

Square Plates — Single Centre Load. 

If a single load P rests at the centre of a square plate, 
using Fig. i and following the same method as in the 
preceding section, the moment about the diagonal AB 
will be: 

7,, Ph sin 45"^ „, , . 

M=- ^ ^^ =.i77Pb. .... (5) 

The moment of inertia I is the same as befoie and it 
is given by eq. (2). Hence, assuming a uniform intensity 
k throughout the extreme fibres : 

,, _ .i7yPbh _sP ,. 

' ii~-4/? ^^^ 

Or, 

h = .S66yj^. (7) 



768 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



Rectangular Plates — Uniform Load. 

Fig. 2 shows a rectangular plate with sides a and h. 
With a much oblong rectangle the indications of tests are 
not so well defined as to the section of failure, but tenta- 
tively the diagonal section AB may be taken as a close 
approximation for usual proportions. DF is a normal 
to AB drawn from D. The uniform load on the triangular 

half ABD of the plate is - — and its centre of action is at 

2 

n 
the normal distance - from AB. The centre of the sup- 

3 
porting forces or reaction along the edges AD and DB is 

- from AB. Hence the moment about AB is 



M 



pah In n 

2 \2 3 



pabn 



12 



(8) 




Fig. 2. 
Referring to Fig. (2) : 

n=b sin (f) and AB =b sec <]). . . (9) 
Therefore the moment of inertia of the diagonal section 



is 



J _b sec (t)h^ ^ . pab^ sin </> _ .b sec (ph^ 



12 



12 



Art. 123.] 
Hence, 



CIRCULAR PLATES. 



769 



7 ah sin </> cos 
k=p -— -\ or 



P 



sin cos 0. (10) 



2/r ' \2k 

As is obvious, P=pab is the total load on the plate. 

Rectangular Plate — Centre Load. 

If a single load P rests at the centre of the plate, the 
moment about the diagonal AB, Fig. 2, is produced by the 
reaction, only, of the supporting forces along the edges 
AB and BD, and its value is 

2 2 

Consequently, 



(11) 



k = 



3P sin cos 4> 



or, h 



-4 



iP 



sm cos 0. 



(12) 



Circular Plate — Uniform Load — Centre Load. 

The circular plate with radius r is shown in Fig. 3. 
The same general assumptions are made as in the preceding 
cases, i.e., uniform condition of 
bending stress throughout the 
section of failure and uniform 
support along the edge of the 
plate. It is clear that the latter 
assumption is strictly correct for 
the circular outline. Any diam- 
eter, as AB, may be taken for 
the section of failure. 

It will be convenient to sup- 
pose the uniform load to be ap- 
plied on a circle of radius ri, as shown in Fig. 3. Then 

the load on half of the plate is p and its centre is at 




770 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

AT 1 

the distance - — from AB. The edge-supporting force or 
reaction, equal to the half load on the plate, has its centre 

2T 

at the distance of — from AB. The moment about the 

TT 

latter diameter is, therefore, 

2 \ TT 37r/ \ 3 / 

If h is the thickness of the plate, as in the preceding 
cases, the moment of inertia 7 is 

r 2rh^ rh^ , . 

1 = =— -; . . , , o , . (14) 

12 o 



Hence, 



Or, 



M=prMr~ri)=k— (15) 

\ 3 / 3 

k=pri^ U • •••... (16) 



h=ri 



VK-'t) <■'> 



If the load is uniform over the entire circular plate, 
r=ri, and 

M = i--; ^ = ^J; and, h = r^^. . . (18) 

If the load is concentrated at essentially a point, ri =0, 

but 2^^Il- must be displaced by — ; 



-^ ' i|; and, h=M. 

irh^ \ irk 

These formulae for circular plates are more nearly 



M=P-\ ^=^; and, h=J^. . . (19) 

TT irh^ \ irk 



Art. 123.] 



ELLIPTICAL PLATES. 



771 



correct in analysis and give results more nearly in agree- 
ment with tests than those derived for other cases. 



Elliptical Plates — Centre Load — Uniform Load. 

An elliptical plate is shown in Fig. 4. The approximate 
formulae for this case may be conveniently established by 
first considering two axial strips of 
the same (unit) width, the length 
of AB being 2a and of CD, 2b, a 
single load being placed at their 
intersection. The centre deflec- 
tions of the tw^o strips as parts of 
the plates must be the same. Let 
Pi be the centre load for the strip 
AB, and P2 the centre load for CD. 

The desired centre deflection for each strip acting as a beam 
is given by eq. (28), Art. 28. The equality of the two de- 
fl.ections gives the equation, 2a being one span and 2b the 
other : 

Pia^ P2b\ _ Pi^b^ 
P2~a^' 




Fig. 4. 



6EI 6EI 



or 



(20) 



h^ 



As each strip is of unit width, I = — , h being the thick- 

12 



ness of plate. Hence the greatest fibre stresses are 

b 



. Mh „ a 



and, k2=zP'. 



h^' 



Eqs. (21) and (20) then give: 

ki _Pi a 
k'2~¥2~b 



62 



(21) 



(22) 



Eq. (22) shows that ^2 is the greatest fibre stress and, 
hence, that the major axis of the ellipse will be the line of 
failure, as would be anticipated without the analysis. 



772 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

If the ellipse of Fig. 4 be elongated by lengthening 
the major axis 2a to infinity, the result will be a corre- 
spondingly long rectangular plate of 26 width or span. 
Hence, the greatest fibre stress for this case of uniform 
load will be for a unit cross strip of plate : 

, Mh p{2hY hi2 ^b^ 

This is the greatest intensity of stress for an ellipse 
whose major axis 2a is infinity. The other extreme is the 
circle for which the greatest intensity of stress is, eq. (18), 

^=^g • • • (24) 

For ellipses in general, in the absence of a satisfactory 
analysis, it is tentatively proposed to wTite: 

-(3-!)f («) 

When b=a, eq. (25) gives the correct value for a circle, 
and when ^ = o the result is correct for the extreme ellipse. 
The thickness of plate for a given uniform load p is 



= ^V(3 -._')! (.6) 



a/ k 

Flat Plates Fixed at Edges. 

Grashof and others have partly by analysis and partly 
empirically deduced a number of formulae for plates fixed 
at their edges, i.e., encastre, instead of simply supported. 
The following have been used and may be considered fairly 
satisfactory, using the same notation as in the preceding 
parts of this article. 

I. Circular plate with radius r and uniform load p. 
The greatest intensity of stress is, if h is the thickness, 



Art. 123.] PLATES WITH FIXED EDGES, 773 

^=^Jp; and, ^='-Jj|- • • • (27) 

II. Stayed flat surfaces, stay bolts being the distance 
c apart in two directions at right angles to each other. 
Each stay carries the uniform load pc^. The greatest 
intensity of stress may be taken: 

k=p-^-, and, h=-J^. . . . (28) 

III. Rectangular plate a long, b wide, supporting uniform 
unit load p. The greatest intensity of stress may be taken : 



k=pTnT-r^^h:^ and, h=a^h^^-j-^y^^. (29) 






If the plate is square, a = b'. 

k=p^,- and, /.=-^^|. . . . (30) 

All these plates with edges either fixed or simply sup- 
ported are supposed to be truly fiat, as any arching or 
dishing changes materially the conditions of stress. 

Problem i. — What thickness of steel plate is required 
to carry a load of 200 pounds per square inch over a rect- 
angular opening 24 by 36 inches. Eq. (10) gives the 
expression for the thickness h of the plate when simpl}^ 
supported along its edges. The total load isP = 2ooX36x 
24 = 168,800 pounds. 

tan (/) =11 = .667 .•. =33° 40' and sin cos (^ = .461. 
If the working stress ^ = 16,000 pounds per square inch; 

h=\— — '- X. 461 =1.56 inches. A plate iye inches 

\2 X 16,000 

thick, therefore, meets the requirements. 

Problem 2. — Design a circular steel plate, simply sup- 
ported on its edge, for an opening 30 inches in diameter 



774 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

to carry a load of loo pounds per square inch, if k =-15,000 
pounds per square inch. r = i5 inches and P = iooXTrr^ = 
100 X 706.9 = 70,690 pounds. 

Eq. (18) then gives: h = is\ = 1.22 inches. A 

>i 1500 

plate I J inches thick will therefore be satisfactory. 

If the plate were rigidly fixed along its edge^ eq. (27) 

shows that the thickness would be: h = i. 22V^ = i inch 

thick. 

Art. 124. Resistance of Flues to Collapse. 

If a circular tube or flue be subjected to external normal 
pressure, such as that of steam or water, the material of 
which it is made will be subjected to compression around 
the tube, in a plane normal to its axis. If the following 
notation be adopted, 

/ = length of tube; 
d = diameter of tube ; 

t = thickness of wall of the tube ; 
p = intensity of excess of external pressure over internal ; 

then will any longitudinal section It, of one side of the tube, 

pld 
be subjected to the pressure — . But. let a unit only of 

length of tube be considered. This portion of the tube is 
approximately in the condition of a column whose length 
and cross-section, respectively, are nd and t. 

The ultimate resistance of such a column is (Art. 35) 

As this ideal column is of rectangular section, 

12 



Art. 124.] RESISTANCE OF FLUES TC COLLAPSE. yji 

and 

But P=pd, hence 



i2d'' 



(i) 



is the greatest intensity of external pressure which the tube 
can carry. But the formulae of Art. 35 are not strictly 
applicable to this ideal column. The curvature on the one 
hand and the pressure on the other tend to keep it in position 
long after it would fail as a column without lateral support. 
Hence p will vary inversely as some power of d much less than 
the third. 

Again, it is clear that a very long tube will be much more 
apt to collapse at its middle portion than a short one, as the 
latter will derive more support from the end attachments; 
and this result has been established by many experiments. 
Hence p must be considered as some inverse function of the 
length /. 

Eq. (i), therefore, can only be taken as typical in form, 
and as showing in a general way, only, how the variable 
elements enter the value of p. If x, y, and z, therefore, are 
variable exponents to be determined by experiment, there 
may be written 

f='M^ (^) 

in which c is an empirical coefficient. 

Sir Wm. Fairbairn (" Useful Information for Engineers, 
Second Series") made many experiments on wrought-iron 
tubes with lap- and butt-joints single riveted. He inferred 



776 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

from his tests that y=z = i. Two different experiments 
would then give 

pld = ct-, (3) 

p'Ud'=ct'' (4) 

Hence 

log {pld) =log c-\-x\ogt, 
\og{p'l'd')=\ogc-^x\ogt'\ 

in which "log" means "logarithm." Subtracting one of 
these last equations from the other, the value of x becomes 

, pld 
log' 



_ log ipld) - log {p'Vd') --S \p'Vd', 

^ \ogt-\ogf ~ /t^ ' ' ' ^^^ 

log[j^ 

As p, I, d, t, p',V,d\ and f are known numerical quantities 
in every pair of tests, x can at once be computed by eq. (5) ; 
c then immediately results from either eq. (3) or eq. (4). 
By the application of these equations to his experimental 
data, Fairbairn found for wrought-iron tubes: 



^ = 9,675,600-^, (6) 



in which p is in pounds per square inch, while t, I, and d are 
in inches. Eq. (6) is only to be applied to lengths between 18 
and 120 inches. 

He also found that the following formula gave results 
agreeing more nearly with those of experiment, though it is 
less simple: 

/^■'9 d 
;^ = 9, 675, 600-^-0. oo2y (7) 



Art. 124.] RESISTANCE OF FLUES TO COLLAPSE. 777 

Fairbairn found that by encircling the tubes with stiff 
rings he increased their resistance to collapse. In cases 
where stick rings exist, it is only necessary to take for I tlie 
distance between two adjacent ones. 

In .1875 Prof. Unwin, who was Fairbairn' s assistant in 
his experimental work, estabHshed formulae with other 
exponents and coefficients (" Proc. Inst, of Civ. Engrs.," 
Vol. XL\^I). He considered x, y, and z variable, and 
found for lubes with a longitudinal lap-joint: 

t-' 
/? = 7,363,000^^^^^6 (8) 

From one tvibe with a longitudinal butt-joint, he deduced: 

/2.21 
^ = 9>6i4,ooo ^o.9^x.x6 .(9) 

For five tubes with longitudinal and circumferential joints^ 
he found: 

:^ = 15, 547, 000^57^7776 (10) 

By using these same experiments of Fairbairn, other 
writers have deduced other formula, which, however, are 
of the same general form as those given above. It is proba- 
ble that the following, which was deduced by J. W. Nystrom, 
will give more satisfactory results than any other: 

^.692,800^ (") 

At the same time, it has the great merit of more simple 
application. 

From one experiment on an elliptical tube, by Fairbairn, 
it would appear that the formulas just given can be approxi- 



778 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI . 



mately applied to such tubes by substituting for d twice 
the radius of curvature of the elliptical section at either 
extremity of the smaller axis. If the greater diameter or 
axis of the ellipse is a and the less 6, then, for d, there is 

to be substituted -r. 



Art. 125. — Approximate Treatment of Solid Metallic Rollers. 

An approximate expression for the resistance of a roller 
may easily be written. The approximation may be con- 
sidered a loose one, but it furnishes a basis for an accurate 
empirical formula. 

The following investigation contains the improvements 
by Prof. J. B. Johnson and Prof. H. T. Eddy on the 

method originally given by the 
author. 

The roller will be assumed 
to be composed of indefinitely 
thin vertical slices parallel to 
its axis. It will also be as- 
sumed that the layers or slices 
act independently of each 
other. 

Let E' be the coefficient of 
elasticity of the metal pver the 
roller. 

Let E be the coefficient of elasticity of the metal of 
the roller. 

Let R be the radius of the roller and R^ the thickness 
of the metal above it. , 




Let ze; = intensity of pressure at A ; 

. (( ({ (( a 



at /i; 

'* any other point 



Art. 125.] TREATMENT OF SOLID METALLIC ROLLERS. 779 

Let P = total weight which the roller sustains per unit 
of length. 
X be measured horizontally from A as the origin ; 
d=AC; 
e = DC, 



From Fig. i : 



E E 

•. d = AC = AB + BC = w{^^-^ . . . (i) 



and 



A'C=A'B^^B'C=p(^^~}j, ... (2) 
Dividing eq. (2) by eq. (i), 



But 



P = jydxJI^£'A'C'dx. 

If the curve BAR be assumed to be a parabola, as may 
be done without essential error, there will result: 



C^' A'Cdx^ 



^ed. 



Hence 



P=-we (3) 



78o 
But 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



e = V2Rd-d'^ = ViRd, nearly. 



By inserting the value of d from eq. (i) in the value of 
e, just determined, then placing the result in eq. (3), 



liR-^R^ 



p=-^V-^^<i+g) ■ • (4) 



-M^^'^ (5) 



The preceding expressions are for one unit of length. 
If the length of the roller is /, its total resistance is 



Or if R=R', 



P'=P/=J^'2Z.=^(| + g 



P' = 



=^i?/^2« 



,£ + £' 



EE' 



(6) 



(7) 



In ordinary bridge practice eq. (7) is sufficiently near 
for all cases. 

A simple expression for conical rollers may be obtained 
by using eqs. (4) or (5). 

!^ I » 




Fig. 2 



As shown in Fig. 2, let 2 be the distance, parallel to the 
axis, of any section from the apex of the cone ; then con- 



Art. 126.] RESISTANCE TO DRIVING AND DRAWING SPIKES. 781 

sider a portion of the conical roller whose length is dz. ■ Let 
R^ be the radius of the base. The radius of the section 
under consideration will then be 

and the weight it will sustain, ii R^=R\ 
Hence 



Eqs. (6), (7), and (8) give ultimate resistances if w is 
the ultimate intensity of resistance for the roller. 

It is to be observed that the main assumptions on which 
the investigation is based lead to an error on the side of 
safety. 

If for wrought iron, ?i; = 12,000 pounds per square inch, 
and £^=E' = 28,000,000 pounds, eq. (5) gives 

Art. 126. —Resistance to Driving and Drawing Spikes. 

Some very interesting experiments on driving and draw- 
ing rail spikes were made by Mr. A. M. Wellington, C.E., 
and reported by him in the " R. R. Gazette,'* Dec. 17, 1880. 
He experimented with wood both in the natural state and 
after it had been treated by the Thilmeny (sulphate of 
baryta) preserving process. 

" The test -blocks were reduced to a uniform thickness of 
4.5 inches, this thickness being just sufficient to give a full 



782 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



bearing surface to the parallel sides of the spikes when 
driven to the usual depth, and to allow the point of the 
spike to project outwards. It was considered that the be v- 

Table I. 

SPIKES WERE STANDARD: 5.5 INCHES X j'^ INCH. 



Kind of Wood. 



Natural Wood. 



To Driving 
Spike, Pounds 



To Pulling 
Spike, Pounds 



Prepared Wood. 



To Driving 
Spike, Pounds 



To Pulling 
Spike, Pounds. 



Beech 

White oak, green 

Pin oak 

White ash . 

White oak, well seasoned 

Black ash 

Elm 

Chestnut, green 

Soft maple 

Sycamore 

Hemlock 



Mean. 

I 5,521 ) ^'■^ 

5,953 



4,758 



4,606 

4,103! , ^^O 



Mean 
6,638 ( . 

6,469 f ^'553 
4,560 



4,408 
4,868 



[4,638 



lilt\-'^° 



{• 3,260 



2,730 
3,790 

1,996 



Mean, 



7,288 
7,656 



6,117 
4,589 
6,588, g 

5,978 r 



\s.: 



283 



4,4531 
4'4S3i 
4,301 j '*' "*' 
3,38oJ 
4,453 (. 
4,148 j 



4,300 



^:^?U 3.833 



Mean. 



8,873 U^, 
8,2675 ^'4' 



(SpHt) 



3,340 

3,028 

3,300 

3,493J 

4,148 

4,202 



'3,290 
4,175 



2,725 
3,030 



2,877 
1,968 



elled point could add very little to the holding power of the 
Spike, and it was desired to press the spike out again by 
direct pressure after turning the block over. ..." 

The forces exerted in pulling and driving the spikes 
were produced by a lever. A few tests with a hydraulic 
press showed that the friction of the plunger varied from 
about 6 to 18 per cent. The experimental results' are 
given in Table I. 

Some very excellent tests of the holding power of rail- 
road spikes and lag-screws were made by Mr. A. J. Cox, of 
the University of Iowa, during 1891, in the engineering 
laboratory of that institution, the results of which were 



Alt. 126.] RESISTANCE TO DRIVING AND DRAWING SPIKES. 

Table II. 



783 



RESISTANCE OF RAILROAD SPIKES TO PULLING OUT AND 
PRESSING IN. 



Kind of Tie and Spike. 



No. 

of 

Tests. 


Greatest Resistance 
in Pounds. 


Maxi- 
mum. 


Average. 


Mini- 
mum. 


20 
9 

I 

2 

4 
3 

7 
3 
5 
3 

2 
2 
2 
2 


7,700 
6,660 


5,514 
4,936 

i 5,120* 
( 4,460 
\ 4,040* 
( 3,240 
5,843 
6,350 

4,706 
5,807 
5,130 
5,334 

1,140 

1,400 

1,775 

955 


3,500 
3,950 

j. 

h- 

5,290 
5,600 

4,050 
5,720 
4,400 
4,030 

1,040 

1,340 

1,720 

930 




6,580 
6,850 

6,130 
5,950 
5,680 
6,930 

1,240 

1,460 

1,830 

980 



Average 
Resistance 
in Pounds 
per Square 
Inch Sur- 
face of Spikef 



Average 
Resistance 
per Ounce 

of Spike. 



Seasoned White-oak Tie. 

Common spike 

Common spike, i-in. bored hole. 

Common spike, redrawn 

Common spike, -^-in bored hole, | 

redrawn j 

Hill curved spike 

Bayonet spike 

Unseasoned White Oak. 

Common spike 

Common spike, ^-in. bored hole. 

Hill curved spike 

Bayonet spike , 

Unseasoned White Cedar. 

Common spike 

Common spike, -^-in. bored hole 

Hill curved spike 

Bayonet spike , 



643 
575 

520 
378 



548 
716 



133 
162 



664 
595 

537 



632 
934 



567 
740 
555 
784 



137 
169 
192 
140 



PRESSING SPIKES INTO TIES UNDER STEADY PRESSURE OF 
TESTING MACHINE. 



White-oak Ties. 

Curved spike, pressing in. . 
Curved spike, pulling out.. 
Bayonet spike, pressing in. 
Bayonet spike, pulling out. 



2 


7,430 


7,375 


7,320 


2 


6,830 


6,615 


6,400 


2 


6,660 


6,530 


6,400 


2 


4,400 


3,84s 


3,290 



* These values are the first resistance to drawing out. 
in the same holes and redrawn, with the results shown, 
t Wedge surface not considered. 



The spikes were then redriven 



published in the technical journal ("The Transit") of the 
university for September, 1891; they will be found some- 
what rearranged in Tables II and III. Three kinds of 
spikes were used, viz., the common spike (length 5.5 ins., 
0.5625 in. square, weight 8.3 oz.), Hill's curved spike (length 
5.875 ins., weight 9.25 oz.), and the bayonet or grooved 



784 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



Spike (length 5.5 ins., weight 6.8 oz.). The timber of the 
ties is shown in the two tables. The spikes were forced 



Table III. 
RESISTANCE OF LAG-SCREWS TO PULLING OUT.* 





Diameter 


Diameter 


Length 


Maximum 


Resistance 


No 


Kind, of Wood. 


of 


of Bored 


Screw 


Average 


pounds 


of 




Screw, 


Hole, 


in Hole, 


Resistance 


per Square 






Inches. 


Inches. 


Ins. 


in Pounds. 


Inch. 




Seasoned white oak. . . . 


% 


V2 


4y2 


8,037 


1,024 


3 


Seasoned white oak. . . . 


%6 


Vie 


3 


6,480 


1,223 


I 


Seasoned white oak. . . . 


V2 


• % 


4y2 


8,780 


1.239 


2 


Yellow-pine stick 


% 


V2 


4 


.3,800 


484 


2 


White cedar unseasoned 


% 


1/2 


4 


3,405 


434 


2 



* The area of surface for these lag-screws used, in finding the resistance per square inch 
was computed as that of a cylinder whose diameter was equal to the diameter of the screw 
considered. In pulling the first lag-screw of Table III, the resistance of 8037 pounds at 
the end of a i-inch movement decreased to 4550, 2476, 1475, and 410 pounds at the ends 
of movements of 0.5, i, 2, and 2.75 inches respectively. 

into the wood by the pressure exerted by the 100,000-potind 
testing machine used in the tests, and by Avhich they were 
pulled out of the ties. 

The greatest pulling resistance of any spike is offered 
at the very beginning of motion, and it then rapidly de- 
creases. A common spike which resisted 5120 pounds at 
the beginning of motion offered but 3050 pounds after 
having moved a half -inch, 2,440 pounds after i inch of mo- 
tion, 1,300 pounds after 1.75 inches, 940 pounds after 2 
inches, and 440 pounds after moving 3 inches; the original 
penetration of the spike was 4.375 inches in a seasoned white- 
oak tie. Similar results were reached with other timbers. 

AVhen spikes were pressed into the ties the timber 
offered an increasing resistance to penetration, but at a 
rate less rapid than that of the decrease in pulling out. A 
^-inch penetration in a seasoned white-oak tie gave a re- 
sistance to a common spike of 2,320 pounds which increased 
to 3,340 pounds for i-inch penetration, to 4550 pounds for 



Art. 126.1 RESISTANCE TO DRIVING AND DRAWING SPIKES. 



785 



2 inches, to 5580 pounds for 3.5 inches, and to 6555 pounds 
for 4.5 inches. 

The following results showing the relative holding 
power of common and screw railroad spikes were found 
by tests made by Prof. W. Kendrick Hatt for the U. S. 
Dept. of Agriculture and published in Forest Service 
Circular 46, 1906. 



Table IV. 

HOLDING FORCE OF COMMON AND SCREW SPIKES. 



Species of Wood and 


Num- 
ber of 
Tests. 


Condition of Wood. 


Force Required to Pull Spike. 


Kind of Spike. 


Average. 


Max. 


Min. 


White oak: 

Common spike 

Screw spike 

Ratio 


5 

5 


Partially seasoned . . 
Partially seasoned . . 


Pounds. 

6,950 
13,026 

1.88 


Pounds. 

7,870 

14,940 


Pounds. 

6,160 

11,050 


Oak (probably red) : 

Common spike 

Screw spike 

Ratio 


5 
8 




Seasoned 

Seasoned 


4,342 

11,240 

2.61 


5,300 
13,530 


3,490 
8,900 




Loblolly pine: 

Common spike 

Screw spike 

Ratio 


28 

26 


Seasoned 


3,670 

7,748 

2. II 


6,000 
14,680 


2,320 
4,170 


Seasoned 


Hardy catalpa: 

Common spike 

Screw spike 

Ratio 


12 

14 



Green. 


3,224 

8,261 

2.56 


4,000 
9,440 


2,190 
6,280 


Green 




Common catalpa: 

Common spike 

Screw spike 

Ratio 


II 
II 


Green. 


2,887 

6,939 
2.42 


4,500 
8,340 


2,240 
5,890 


Green 




Chestnut: 

Common spike 


4 

5 


Seasoned 


1 

2,980 

9,418 

3-15 


3,220 
11,150 




Screw spike 


Seasoned 


7,470 


Ratio 















786 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



Table V. 

HOLDING FORCE OF COMMON AND SCREW SPIKES. 
Seasoned Clear and Knotty Loblolly Pine Ties. 



Position of Spike. 


Kind of Spike. 


Number 
of Tests. 


Force Required to Pull Spike, 


Average. 


Max. 


Min. 


In clear wood 


Common 

Common 

Screw 


36 
18 
40 
20 


Pounds. 
3,466 
2,615 
7,180 
9,763 


Pounds. 

6,250 

3,750 

13,710 

17,200 


Pounds. 
I 880 


In knotty wood 


1,010 
2,000 
4,890 


In knotty wood 


Screw 





Art. 127. — Shearing Resistance of Timber behind Bolt or Mortise 

Holes. 

Col. T. T. S. Laidley, U.S.A., made some tests during 
1 88 1 at the United States Arsenal, Watertown, Mass., on 
the resistance offered by timber to the shearing out of bolts 
or keys when the force is exerted parallel to the fibres. 




Fig. I. 



Fig. 2. 



The test specimens are shown in Figs, i and 2. Wrought- 
iron bolts and square wrought-iron keys were used. All 
the timber specimens were six inches wide and two inches 
thick. The diameter of the bolts used (Fig. i) was one 
inch for all the specimens. The keys were i''Xi.''5 and 
I'^I25XI'^5 as shown in Fig. 2. In all the latter speci- 
mens, failure took place in front of the smaller key where 
the pressure was greatest. 



Art. 127. 



SHEARING RESISTANCE OF TIMBER. 



787 



In many cases the specimen sheared and split simultane- 
ously in front of the hole. By putting bolts through the 
pieces in a direction normal to the force exerted, so as to 
prevent splitting, the resistance was found (in most cases) 
to be- considerably, though irregularly, increased. 

Unless otherwise stated, the wood was thoroughly sea- 
soned. 

The accompanying table gives the results of Col. Laid- 
lev's tests. 



Kind of Wood. 


Centre 
of Hole 
from 
End of 
Speci- 
men. 


Total 

Area 

of 

Shearing. 


Ultimate Shearing Resistance 
per Square Inch, in Pounds. 


Spruce (bolts) 


Ins. 

li 

18 


Sq. Ins. 

8 

16 

24 

32 


399 
359 




275 
202 




P 


8 


457 


White pine (bolts) 


- 


4 
6 

18 


16 

24 
32 


611 
450 

327 


Yellow pine (bolts) 




f2 

4 
6 

18 


8 
16 
24 
32 


607 
720 
456 

337 






'2 


8 


599 


Yellow pine (square keys) . 


■' 


4 
6 


16 

24 


369 

572 




[7 


28 


438 




r 2 


8 


550 


White pine (square keys) . 


J 4 

17 


16 

24 ■ 

28 


412 
332 
236 






' 2 


8 


410 (Not thoroughly seasoned.) 


Spruce (square keys) 




4 
6 

l7 


16 

24 

28 


329 " " " 
242 (Wet timber.) 

279 



788 MISCELLANEOUS SUBJECTS. [Ch. XVL 

Art. 128. — Method of Least Work — Stresses in a Bridge Portal. 

In the consideration of stresses in structures or parts of 
structures where the equations of condition for statical 
equilibrium are not enough to determine all the unknown 
quantities, it is necessary to find other equations involving 
the elastic properties of the materials used. The Method 
of Least Work affords one. procedure by which such extra 
equations may be found. 

If a force P is gradually applied at a point in a struc- 
ture it produces a deflection or distortion 5 in its own 
direction and performs the work, 

W^=^=^=^. ..... (I) 

2 2 2a 

As a consequence of Hook's law P=ad, a being a con- 
stant and a direct function of the modulus of elasticity 
E or G. Hence 

This is called the first theorem of Castigliano, enun- 
ciated in his " Theorie des Gleichgewichtes elastischer 
Systeme." Eq. (2) is perfectl}^ general and includes all 
elastic deformation or deflection. It shows that the first 
derivative of W, the work performed by the load P, in 
respect to that load as the independent variable, is the 
elastic distortion as well in the case of a force acting axially 
along a bar either in tension or compression as in that of a 
load producing deflection of a bridge at its point of applica- 
tion. 

The third member of eq. (i) shows that 

dW , ^ .X 

---=a8=P (3) 



Art. 128.] STRESSES IN A BRIDGE PORTAL. 789 

This equation may at times be useful. 

If the point of appHcation of the force or load P in eq. 
(2) be supposed unchanged in position while P acts, the 
other parts of the structure or piece moving in adjust- 
ment to that condition as may be required by the corre- 
sponding strains, then will 5 =0 and 



dW 
dP 



(4) 



If this equation be satisfied by solving it for P, the 
resulting value will make W, in general, either a maxi- 
mum or minimum. In engineering structures, however, 
it is obvious that W will be a minimum, as the test by the 
second derivative will show in individual cases. 

Eq. (4) expresses Castigliano's second theorem. If 
then the first derivative of a function W expressing the 
work performed in distorting a structure or structural 
member in terms of an indeterminate force or stress P, 
whose point of application may be supposed fixed, be taken 
in reference to that indeterminate force as the variable, 
a new equation of condition will result whose solution 
will yield a value of the force making the energy expended 
in the elastic distortions the least possible. Hence this 
procedure is called " the method of least work." 

Stresses in a Bridge Portal. 

The treatment of a bridge portal will illustrate the use 
of the method of least work in treating an important 
part of a bridge. Fig. i shows a skeleton diagram of the 
portal, AF and BG being the end posts in full length h in 
their own plane. ABCD is the outline of the portal brac- 
ing which may be a plate girder or open bracing. The 
corner or gusset bracings at C and D are omitted. The 



790 



MISCELLANEOUS SUBJECTS. 



[Ch. XVI. 



equal end post stresses due to vertical dead and moving 
loads are indicated by P and P. The total horizontal 
wind load acting at the upper ends of the end posts is shown 
by //, and it is taken as applied wholly on the windward 
side. As is usual, the end posts are considered fixed in 
direction at both upper and lower ends. The lateral 




Fig. I. 



action of the wind will distort the portal in the manner 
shown exaggerated. 

As both posts are supposed to be in the same condition 
and equally affected by the lateral wind pressure, the two 
points of contrafiexure K and must be at same distance 
ho (to be determined) from FQ. The points M' and M'^ 
are in the neutral surface of ABCD, i.e., at its mid-depth. 
The notation of Fig. i is self explanatory. The left arrow 

TT 

— is below K and external to the upper part of Fig. i, 



Art. 128.] STRESSES IN A BRIDGE PORTAL. 791 

M 

but the right arrow — is above and external to the 

2 

lower part of the figure. Right-hand moments are posi- 
tive and left-hand negative. Taking moments of forces 
acting on the upper part ABOK of the portal and about 0\ 

{P,-P=P')b-H{h-ho)=o .'. P'b=H{h-ho). (5) 

Obviously, P' is the transferred load from the wind- 
ward truss to the leeward due to the wind pressure H. 

In order to find the work performed in distorting the 
members of the portal, it is necessary to determine the 
bending moments M\ M" , M2 and Mi at the points indi- 
cated by these letters. Taking a section through K and 
moments about M', Fig. i : 



M'=--(h-ho--) -H- + (P'b=H{h-ho)) 
2 \ 2/2 

... M'=^(h-ho-^) (6) 

Then moments about ilf will give 

M''=-^(h-ho-fj = -M\ ... (7) 



Obviously the signs of the moments M2 and Mi must 
be opposite to those of M" and M^ respectively. Hence, 

M2=~ho; and Mi = --ho. ... (8) 
2 2 



The moments throughout the parts of the portal will 
then be: 



792 MISCELLANEOUS SUBJECTS. [Ch. XVI. 

For girder AC, 

For left post FA, 

«.=«.-^^-?('-('-3f)- ■ (.") 

For right post BG, 

It has been shown in the chapter on resilience that the 

work done in bending a beam is ^=ry ( M^dx, I being 

the moment of inertia of the normal section of the beam. 
Similarly the work performed by an axial force P on a 
straight member whose area of cross-section is A and 

length h is — —r^. If Ii is the moment of inertia for the 
2AE 

member AC, Fig. i, and 1 2 for each post AF and BG, while 

A 2 is the common area of cross-section for the latter, one 

TT 

carrying the axial load P-\- — {h—ho) and the other 

TT 

P — r-(^~^o), the total work done on the entire portal is 


2EI1J0 2EI2J0 



Art. 128.] STRESSES IN A BRIDGE PORTAL, 793 

d H%^ 

If n=h — and g =P'^-] — t-t— there will result: 
2 0^ 



w^ ^'' 



24EI 



//2/j /^2 \ 

- (n2 - 2nho +/io^) H — f^-( hon +ho'^ I 

1 4^/2X3 / 



2H% 



/io+^V). (12) 



' A2EV b^ 62 

Eq. (5) shows that ho may be replaced in this equation 

in terms of P'; hence -r— corresponds to -p=-,. Placing 

dho dP 

-r— = o and solving, therefore, 

a/zo 



h --] (bl2 +3hIi)b^A2+24h^Ij2 



b^A2l2+6hb^A2li+24hIiL. 



(13) 



This locates the points of contraflexure and enables 
all computations to be made. 

If the axial compression of the two end posts be neglected, 
the last term in both numerator and denominator of the 
second member of eq. (13) disappears, and 



_{^-f) 



{hU+ihh) 



^= — bu+m, — ^'4) 

If r2 is the radius of gyration of yl2 and if -y =i, eq. 
(13) may take a more convenient form for computation: 



( I --TJ(6^*+3^)^ + 24r22- 






794 MISCELLANEOUS SUBJECTS, [Ch. XVI. 

In the same manner eq. (14) becomes: 



i^-iY^+sh) 



ho=- j^ (16) 



CHAPTER XVII. 
THE FATIGUE OF METALS. 

Art. 129.— Woehler's Law. 

In all the preceding pages, that force or stress which, 
by a single or gradual application, will cause the failure or 
rupture of a piece of material has been called its *' ultimate 
resistance." It has long been known, however, that a stress 
less than the ultimate resistance may cause rupture if its 
application be repeated (without shock) a sufficient number 
of times. Preceding 1859 no experiments had been made 
for the purpose of esxablishing any law connecting the num- 
ber of applications with the stress requisite for rupture, or 
with the variation between the greatest and least values of 
the applied stress. 

During the interval between 1859 and 1870, A. Wohler, 
under the auspices of the Prussian Government, undertook 
the execution of some experiments, at the completion of 
which he had established the following law : 

Rupture may he caused not only by a force which exceeds 
the ultimate resistance, but by the repeated action of forces 
alternately rising and falling between certain limits, the greater 
of which is less than the ultimate resistance; the number of 
repetitions requisite for rupture being an inverse function 
both of this variation of the applied force and its upper limit. 

This phenomenon of the decrease in value of the break- 

795 



796 



THE FATIGUE OF METALS. 



fCh. XVII. 



ing load with an increase of repetitions is known as *Hhe 

fatigue of materials.'' 

Although the experimental work requisite to give 
Wohler's law complete quantitative expression in the 
various conditions of engineering constructions can scarcely 
be considered more than begun, yet enough has been done 
by Wohler and Spangenberg to establish the fact of metallic 
fatigue, and a few simple formulae, provisional though they 
may be. The importance of the subject in its relation to 
the durability of all iron and steel structures is of such a 
high character that a synopsis of some of the experimental 
results of Wohler and Spangenberg will be given in the next 
article. 



Art. 130. — Experimental Results. 

The experiments of Wohler are given in '' Zeitschrift fiir 
Bauwesen," Vols. X., XIII., XVI., and XX., and those of 
Spangenberg may be consulted in "Fatigue of Metals," 
translated from the German of Prof. Ludwig Spangenberg, 
1876. 

These results show in a very marked manner the effect 
of repeated vibrations on the intensity of stress required 
to produce rupture. 

Spangenberg states that "the experiments show that vi- 
brations may take place between the following limits with 
equal security against rupture by tearing or crushing: 

r + 17,600 and- 17,600 lbs. per sq. in. 
Wrought iron -I + 33,ooo and— o 

L +48,400 and +26,400 

f +30,800 and— 30,800 
Axle cast steel ] + 52,800 and o 

I +88,000 and+ 38,500 

r +55,000 and o 

I +77,000 and+ 27,500 
Spring-steel not hardened. . i ^§8,000 and + 44.000 

[ +99,000 and+ 66,000 



Art. 130.] EXPERIMENTAL RESULTS. 

And for axle cast steel in shearing : 

+ 24,200 and — 24,200 lbs. per sq. in. 
+ 41.800 and o " " " " 



797 



PHCENIX IRON IN TENSION. 



Pounds Stress per 
Square Inch. 


Number 
of Repetitions. 


Pounds Stress per 
Square Ihch. 


Number 
of Repetitions. 


to 52,800 
to 48,400 
to 44,000 
to 39.600 


800 rupture 
106,910 rupture 
340,853 rupture 
409,481 rupture 


to 39.600 

to 35.200 

22,000 to 48,400 

26,400 to 48,400 


480,852 rupture 
10,141,645 rupture 
2,373.424 rupture 
4,000,000 not broken 



WESTPHALIA IRON IN TENSION. 



to 52,800 


4,700 rupture 


to 39,600 


180,800 rupture 


to 48,400 


83,199 rupture 


to 39,600 


596,089 rupture 


to 48,400 


33,230 rupture 


to 39,600 


433.572 rupture 


to 44,000 


136,700 rupture 


to 35,200 


280,121 rupture 


to 44,000 


159.639 rupture 


to 35,200 


566,344 rupture 



FIRTH & SONS' STEEL IN TENSION. 



to 66,000 


83,319 rupture 


to 55,000 


103,540 rupture 


to 60,500 


168,396 rupture 


to 53,900 


12,200,000 not broken 


to 55. 000 


133,910 rupture 


to 53,900 


229,230 rupture 


to 55.000 


185,680 rupture 


to 52,800 


692,543 rupture 


to 55.000 


360,23s rupture 


to 52,800 


12,200,000 not broken 


to 55.000 


186,005 rupture 


to 50,600 





KRUPP'S AXLE-STEEL IN TENSION. 



to 88 


000 


18 


741 


rupture 


to 


55 


000 


473.766 


rupture 


to 77 


000 


46 


286 


rupture 


to 


52 


800 


13,600 


000 


not broken 


to 66 


000 


170 


000 


rupture 


to 


50 


600 


12,200 


000 


not broken 


to 60 


500 


123,770 


rupture 












~ 



PHOSPHOR-BRONZE (UNWORKED) IN TENSION 



o to 27,500 
o to 22,000 
o to 16,500 



147,850 rupture 

408,350 rupture 

2,731,161 rupture 



o to 13,750 
o to 13.750 



1,548,920 rupture 
2,340,000 rupture 



PHOSPHOR-BRONZE (WROUGHT) IN TENSION. 



o to 22,000 
o to 16,500 



53,900 rupture 
2,600,000 not broken 




,621,300 rupture 



798 



THE FATIGUE OF METALS. 
COMMON BRONZE IN TENSION. 



[Ch. XVII. 



o to 22.000 
o to 16,500 



4 200 rupture 
6,300 rupture 



o to 11,000 



5,447,600 rupture 



PHCENIX IRON IN FLEXURE (ONE DIRECTION ONLY). 



Pounds Stress per 
Square Inch. 


Number 
of Repetitions. 


Pounds Stress per 
Square Inch. 


Number 
of Repetitions. 


to 60.500 
to 55,000 
to 40.500 
to 44,000 


169,750 rupture 

420,000 rupture 

481,97s rupture 

1,320,000 rupture 


to 39,600 
to 35.200 
to 33.000 


4.035,400 rupture 
3,420,000 rupture 
4,820,000 not broken 



WESTPHALIA IRON IN FLEXURE (ONE DIRECTION ONLY). 



o to 52.250 
o to 49,500 
o to 46.750 



612,065 rupture 
457,229 rupture 
799,543 rupture 



o to 44.000 
o to 39,600 



1,493.511 rupture 
3.587,509 rupture 



HOMOGENEOUS IRON IN FLEXURE (ONE DIRECTION ONLY). 



o to 60,500 
o to 55,000 
o to 49,500 
o to 44 000 



169,750 rupture 

420,000 rupture 

481,975 rupture 

1,320,000 rupture 



o to 39,600 
o to 35,020 
o to 33,000 



4,035,400 rupture 
3,420,000 not broken 
48,200,000 not broken 



FIRTH & SONS' STEEL IN FLEXURE (ONE DIRECTION ONLY). 



o to 63,250 
o to 60,500 
o to 55, 000 



281,856 rupture 
266,556 rupture 
,479,908 rupture 



o to 52,250 
o to 49,50c 
o to 49,500 



578,323 rupture 

5,640,596* rupture 

13,700.000 not broken 



♦Accidental. 



KRUPP'S AXLE-STEEL IN FLEXURE (ONE DIRECTION ONLY). 



o to 77 000 
o to 66.000 
c to 60,500 



104,300 rupture 
317,275 rupture 
612,500 rupture 



o to 55.000 
o to '5 5, 000 
o to 49,500 



729,400 rupture 
1.499,600 rupture 
43,000,000 not broken- 



Art. 130.]. EXPERIMENTAL RESULTS. 799 

KRUPFS SPRING-STEEL IN FLEXURE (ONE DIRECTION ONLY). 



to 110,000 


39,950 rupture 


72,600 to 110,000 


19.673.300 not broken 


to 88, 000 


1 17,000 rupture 


66,000 to 99,000 


33,600,000 not broken 


to 66,000 


468,200 rupture 


44,000 to 88,000 


35,800,000 not broken 


to ss 000 


40,600,000 not broken 


44,000 to 88,000 


38,000.000 not broken 


to -49.500 


^2, 942, 000 not broken 


6i,6oo to 88,000 


36,000,000 not broken 


88,000 to tJ2,000 


35,600,000 not broken 


27,500 to 77,000 


36,600,000 not broken 


99,000 to 132,000 


33,478,700 not broken 


33,000 to 77,000 


31,152,000 not broken 



PHOSPHOR-BRONZE IN FLEXURE (ONE DIRECTION ONLY). 



Pounds Stress per 
Square Inch. 



o to 22,000 
o to 19,800 



Number 
of Repetitions. 



862,980 rupture 
,151,811 rupture 



Pounds Stress per 
Square Inch. 



o to 
o to 



[6,500 
[3,200 



Number 
of Repetitions. 



5,075,169 rupture 
10,000,000 not broken 



COMMON BRONZE IN FLEXURE (ONE DIRECTION ONLY). 



o to 22,000 
o to 19,800 



102,659 rupture 
151,310 rupture 




837,760 rupture 
10,400,000 not broken 



PHCENIX IRON IN TORSION (BOTH DIRECTIONS). 



— 35,200 to +35,200 

— -(3.000 to + J,^ 000 

— 28,600 to +28 600 

— 26,400 to + 26.400 


56,430 rupture 

99,000 rupture 

479,490 rupture 

909,810 rupture 


— 24,200 to +24,200 

— 22,000 to +22,000 

— 19,800 to + 19,800 

— 17,600 to + 17,600 


3.632,588 rupture 
4.917,992 rupture 
19 186,791 rupture 
132,250,000 not broken 



ENGLISH SPINDLE-IRON IN TORSION (BOTH DIRECTIONS) 



— 37,400 to + 37.40c 


204,400 rupture 


— 30,800 to +30,80' 


979,100 rupture 


-37.400 to +37,400 


147.800 rupture 


-28 600 to +28,60. 


1,142,600 rupture 


— 35,200 to +35,200 


911,100 rupture 


— 28.600 to + 28,600 


595,910 rupture 


— 35,200 to +35. 200 


402,900 rupture 


— 26,400 to +26,400 


3,823,200 rupture 


— 33,000 to +33,000 


1,064,700 rupture 


— 26,400 to + 26,400 


6 100,000 not broken 


— a 000 to +33.000 


384,800 rupture 


— 22,000 to + 22,000 


8,800 000 not broken 


— 30 800 to +30,800 


1,337.700 rupture 


— 22,000 to + 22,000 


4.000,000 not broken 



KRUPP'S AXLE-STEEL IN TORSION (BOTH DIRECTIONS) 



— 44,000 to +44,000 


367,400 rupture 


— 39,600 to +39 60c 


925,800 rupture 


— 37,400 to +37,400 


4,000.000 not broken 


— 35,200 to +35, 200 


4.800,000 not broken 


— 33.000 to +33 000 


5, 000 000 not broken 



46,200 to + 46.20- 
37,400 to + 37,20c 
35 200 to +35,200 
33,000 to + 33,000 
53,000 to +33.000 



55,100 rupture 

7 07,. "^2 5 rupture 

I 665,580 rupture 

4.163 37 5 rupture 

45,050.640 rupture 



8oo THE FATIGUE OF METALS. [Ch. XVII. 

The late Capt. Rodman, U.SA., made a considerable 
number of experiments on the fatigue of cast iron, but they 
were sufficient in number and character to show the general 
effect only, and gave no quantitative results. 

The specimens used in all the preceding experiments 
were small. 

During i860, '61, and '62 Sir Wm. Fairbairn con- 
structed a built beam of plates and angles with a depth of 
16 inches, clear span of 20 feet, and estimated centre break- 
ing load of 26,880 pounds. 

This beam was subjected to the action of a centre load 
of 6643 pounds, alternately applied and relieved eight 
times per minute; 596,790 continuous applications pro- 
duced no visible alterations. 

The load was then increased from one fourth to two 
sevenths the breaking weight, and 403,2 10 more applications 
were made without apparent injury. 

The load was next increased to two fifths the breaking 
weight, or to 10,486 pounds; 5175 changes then broke the 
beam in the tension flange near the centre. 

The total number of applications was thus 1,005,175. 

The beam w^as then repaired and loaded with 10,500 
pounds at centre 158 times, then with 8025 pounds 25,900 
times, and finally with 6643 pounds enough times to make a 
total of 3,150,000. 

In these experiments the load was completely removed 
each time. 

It is thus seen that vibrations (without shock) with one- 
fourth the calculated breaking centre load produced no 
apparent effect on the resistance of the beam, but that 
two fifths of that load caused failure after a comparatively 
sm.all number of repetitions. 

It is probable that the breaking centre load w^as calcu- 



Art. 131.] FORMUL/E OF LAUNHARDT AND IVEYRAUCH. 801 

lated too high, in which case the ratios i and f should be 
somewhat increased. 

Art. 131. — Formulae of Launhardt and Weyrauch. 

Let R represent the intensity (stress per square unit ot 
section) of ultimate resistance for any material in tension, 
compression, shearing, torsion, or bending; R will cause rup- 
ture at a single, gradual application. But the material may 
also be. ruptured if it is subjected a sufficient number of 
times, and alternately, to the intensities P and Q, Q being 
less than P and both less than R, while all are of the same 
kind. AVhen Q = o let P = IF, and let D = P - Q. IF is called 
the "primitive safe resistance," since the bar returns to its 
primitive unstressed condition at each application. In the 
general case P is called the ''working ultimate resistance." 

By the notation adopted: 

P = Q + P (i) 

But by Wohler's law, P is a function of D, or 

p-KD). . : (2) 

A sufficient number of experiments have not yet been 
made in order to complete!}^ determine the form of the 
function / (P). 

It is known, however, that 

forQ = o, P=P = IF; 
andforP=o, P=Q=R. 

Provisionally, Launhardt satisfies these two extreme 
conditions by taking 

P = fc-?-fcJ<--» <3) 



8o2 THE FATIGUE OF METALS. [Ch. XVII. 

Even at these limits this is not thoroughly satisfactory, 
for when D^o, P = ~{R-W), or is indeterminate. 

By solving eq. (3), 

P-.r(.+^g). .... a, 

But if the least value of the total stress to which any 
member of a structure is subjected is represented by min By 

and its e^reatest value by max B, there will result 7^ =M. 

maxB P 

Hence 

^ ^'J R-W min B\ , ^ 

which is Launhardt's formula. In the preceding article 
some values of W are shown. In applying eq. (5) it is only 
necessary to take the primitive safe resistance, T^F, for the 
total number of times which the structure will be subjected 
to loads. Since bridges are expected to possess an indefinite 
duration of life, in such structures that number should be 
indefinitely large. 

Eq. (5), it is to be borne in mind, is to be applied when 
the piece is always subjected to stress of one kind, or in one 
direction only. It agrees well with some experiments by 
Wohler on Krupx)'s untempered cast spring steel. 

If the stress in any piece varies from one kind to another, 
as from tension to compression, or vice versa, or from one 
direction to another, as in torsion on each side of a state of 
no stress, Weyrauch has established the following formula 
by a course of reasoning similar to that used by Launhardt. 

If the opposite stresses, which will cause rupture by a 
certain number of applications, are equal in intensity, and 



Art. 131.] FORMUL/E OF LAUNHARDT AND JVEYRAUCH. 803 

if that intensity is represented by 5, then will S be called 
the " vibration resistance" ; this was established by Wohler 
for some cases, and some of its values are given in the pre- 
ceding article. 

Let +P and — P' represent two intensities of opposite 
kinds or in opposite directions, of which P is numerically the 
greater. Then if D =P-\-P\ 

P^D-P\ 

The two following limiting conditions will hold: 

ForP'==o, P=D=W\ 
ForP'=5; P=S = \D. 

But by Wohler's law P=f(D), and the two limiting 
conditions just given will be found to be satisfied by the 
provisional formula 

W-S W-S 

^=^1^353p^=IIF353p(P + n. . = (6) 

By the solution of eq. (6), 

/ W-S P'\ 

If, without regard to kind or direction, max B is numer- 
ically the greatest total stress which the piece has to carry, 
while max B' is the greatest total stress of the other kind 

P' m^ax B^ 

or direction, then will -tt = r^- Hence there will result 

' P max B 

the following, which is the formula of Weyrauch: 



/ W-S max B 
P-=W(i TTf -^] (8) 

V W max B ' ^ ^ 



8o4 THE FATIGUE OF METALS, [Ch. XVII. 

Eqs. (5) and (8) give values of the intensity P which are 
to be used in determining the cross-section of pieces de- 
signed to carry given amounts of stress. If n is the safety 
factor and F the total stress to be carried, the area of sec- 
tion desired will be 

^ - p » 

P . 

in which — is the greatest working stress permitted. 

If for wrought iron in tension IF = 30,000 and R = 
50,000, eq. (5) gives 

^ / 2 min B \ 

P = ^0,000 I -\ ^ • 

^ ' • \ 3 max BJ 

Hence, if the total stress due to fixed and moving loads 
in the web member of a truss is max Z^ = 80,000 pounds, 
while that due to the fixed load alone is min 5 =40,000, 
there will result 

^ / 2 40,ooo\ 

P = ^0,000 I -i- - . a " = 40,000. 

^ ' V 3 80,000/ 

In such a case the grea.test permissible working stress 
with a safety factor of 3 would be about 13,300 pounds. 
For steel in tension, if IF = 50,000 and i^ = 75,000, 

^ / I min B 

P = 50,000 I + 5 

^ \ 2 max B 

For wrought iron in torsion, if 5 -= 18,000 and IF = 24,000, 
eq. (8) will give 

^ / I max B^\ 

P = 24.000 I — - — — ^ |. 
\ 4 max B I 



Art. 132.] INFLUENCE OF TIME ON STRAINS. 805 

Other methods based on Wohler's experiments have been 
deduced by Miiller, Gerber, and Schaffer, of which synopses 
may be found in Du Bois' translation of Weyrauch's 
" Structures of Iron and Steel." 



Art. 132. — Influence of Time on Strains. 

In an earlier section of this book devoted to data of 
certain tests, the effect of prolonged tensile stress and 
subsequent rest between the elastic limit and ultimate resist- 
ance was shown to be the elevation of both those quantities. 
It is a matter of common observation, however, that if a piece 
of wrought- iron be subjected to a tensile stress nearly equal 
to its ultimate resistance, and held in that condition, the 
stretch will increase as the time elapses. 

Experiments are still lacking which may show that a 
piece of metal can be ruptured by a tensile stress m.uch 
below its ultimate resistance. It may be indirectly inferred, 
however, from experiments on flexure, that such failure 
may be produced, as the following by Prof. Thurston will 
show. 

A bar 10 parts tin and 90 parts copper, i X i X 22 inches 
and supported at each end, sustained about 65 per cent, of 
its breaking load at the centre for five minutes. During 
that time its deflection increased 0.021 inch. The same 
bar sustained 1485 pounds at centre for 13 minutes and 
then failed. 

A second bar of the same size, but 90 parts tin and 10 
parts copper, was loaded at the centre with 160 pounds, 
causing a deflection of 1.294 inches. After 10 minutes the 
deflection had increased 0.025 i^^ch ; after one day, i .00 inch ; 
after two days, 2.00 inches ; and after three days, 3.00 inches, 
when the bar failed, under the load of 160 pounds. 

Another bar of the same size show^ed remarkable results ; 



So6 THE FATIGUE OF METALS, 



ICh. XVII 



it was composed of 90 parts zinc and 10 parts copper. It 
gave tlie same general increase of deflection with time, but 
eventually broke under a centre load which ran down from 
1233 to 911 pounds, after holding the latter about three 
minutes. 

A bar of the same size and 96 parts copper with 4 parts 
tin, after it had carried 700 pounds at centre for sixty min- 
utes was loaded with 1000 pounds, with the following 
results : 

After. Deflection. 

o minute 3 . 1 18 inches. 



5 minutes 3 

1 5 minutes 3 

45 minutes 4 

75 minutes . 7 

Broke under 1000 pounds. 



540 
660 
102 
634 



A wrought-iron bar of the same size gave, under a centre 
load of 1600 pounds: 

After. Deflection. 

o minute o . 489 inch. 

3 minutes 0.632 " 

6 minutes o. 650 " 

16 minutes o . 660 " 

344 minutes o . 660 " 

It subsequently carried 2589 pounds with a deflection of 
4.67 inches. 

During 1875 and 1876 Prof. Thurston made a number of 
other similar experiments with the same general results. 

Metals like tin and many of its alloys showed an increas- 
ing rate of deflection and final failure, far below the so-called 
"ultimate resistance." The wrought-iron bars, however, 
showed a decreasing increment of deflection, which finally 
became zero, leaving the deflection constant. 

Whether there may be a point for every metal, beyond 



Art. 132.] INFLUENCE OF TIME ON STRAINS. 807 

which, with a given load, the increment of deflection may- 
retain its value or go on increasing until failure, and below 
which this increment decreases as the time elapses, and 
finally becomes zero, is yet undetermined, but seems proba- 
ble. 

It does not follow, therefore, that the principle enunci- 
ated in the section named at the beginning of this article 
is to be taken without qualification. If "rest" imder 
stress, too near the ultimate resistance, be sufficiently pro- 
longed, it has been seen that it is possible that failure may 
take place. 

In verifying some experimental results by Herman 
Haupt, determined over forty years ago. Prof. Thurston 
tested three seasoned pine beams about i\ inches square 
and 40 inches length of span, and found that 60 per cent, 
of the ordinary "breaking load" caused failure at the end 
of 8, 12, and 15 months. In these cases the deflection slowly 
and steadily increased during the periods named. 

Two other sets of three pine beams each broke under 80 
and 95 per cent, of the usual "breaking load," after much 
shorter intervals of time. 

In all these instances it is evident that the molecules 
under the greatest stress " flow" over each other to a greater 
or less extent. In the cases of decreasing increments of 
strain, the new positions afford capacity of increased resist- 
ance ; in the others, those movements are so great that the 
distances between some of the molecules exceed the reach 
of molecular action, and failure follows. 

In many cases strained portions of material recover par- 
tially or wholly from permanent set. In such cases a por- 
tion of the material has been subjected to intensities of 
stress high enough to produce true " flow" of the molecules, 
while the remaining portion has not. The internal elastic 
stresses in the latter portion, after the removal of the exter- 



8o8 THE FATIGUE OF METALS. [Ch. XVII. 

nal forces,, produce in time a reverse flow in consequence of 
the elastic endeavor to resume the original shape. 

It is altogether probable that the phenomicna of fatigue 
and flow of metals are very intimately associated. Some 
of the prominent characteristics of the latter will be given 
in the next chapteio 



CHAPTER XVIIL 

THE FLOW OF SOLIDS. 

Art. 133. — General Statements. 

Although there is no reason to suppose that true solids 
may not retain a definite shape for an indefinite lengtli of 
time if subjected to no external force other than gravity,"^ 
many phenomena resulting both from direct experiment for 
the purpose, and incidentally from other experiments involv- 
ing the application of external stress of considerable inten- 
sity, show that a proper intensity of internal stress (in 
many cases comparatively low) will cause the molecules of a 
solid to flow at ordinary temperatures like those of a liquid. 
And this flow, moreover, is entirely different from, and inde- 
pendent of, the elastic properties of the material; for it 
arises from a permanent and considerable relative displace- 
ment of the molecules. Nor is it to be confounded with 
that internal ''friction" which, if an elastic body is sub- 
jected to oscillations, causes the amplitudes to gradually 
decrease and finallv disappear, even in vacuo. This latter 
motion is typically elastic and the retarding cause may be 
considered a kind of elastic friction. 

It is evident that if a mass of material be enclosed on ah 
its faces, or outer surfaces, but one or a portion of one, and 
if external pressure be brought to bear on those faces, the 

*This, perhaps, may be considered a definition of a true solid. 



8io 



THE FLO IV OF SOLIDS. 



[Ch. XVIII. 



F G_ 

:x__T___iL_.B- 



« b. 



Fig. 



material will be forced to move to and through the free sur- 
face; in other words, the flow of the material will take place 
in the direction of least resistance. 

The theor\^ of the flow of solids 
to be given is that developed by 
Mons. H. Tresca in his " Memoire 
sur I'Ecoulement des Corps So- 
lides," 1865. He made a large 
number of experiments on hard 
and soft metals, ceramic pastes, 
sand, and shot. 

These different materials all 
manifested the same characteris- 
tics of flow, which are well shown 
in Fig. 2. ABCD, Fig. i, is sup- 
posed to be a cylindrical mass of 
lead with circular horizontal sec- 
tion, confined in a circular cylin- 
der, MN, closed at one end wnth 
the exception of the orifice 0. 

This cylinder is supported on 
the base PA', while the face AB 
of the lead receives external pres- 
sure from a close-fitting piston. 
When the pressure is sufficiently 
increased, the face .45 in Fig. i 
sinks to AB in Fig. 2, while the 
column hkHK, in the latter figure, 
is forced to flow through the ori- 
fice 0. 

In Tresca' s experiments with 
lead, the diameter AB was about 3.9 ^'-.ches; the diameter 
HK of the orifice, from 0.75 in. to 1.5 ins., while the length 
of the column or jet hK varied from 0.4 in, to about 24 ins. 



^ 


A F G B 






r^' r'J-Ill-' 






D h 


lli I'l 


k c 


} 


p 


h 


' 1 

H 


^ 


N 



Fig. 2. 



Fig. 



Art. 134.1 TRESCA'S HYPOTHESES. 811 

The total pressure on the face AB varied from 119,000 to 
198,000 pounds. The mitial thickness AD varied from 0.24 
inch to 2.4 inches. 

Some experiments exhibiting in a remarkably clear man- 
ner the flow of metals in cold punching were made by David 
Townsend in 1878, and the results were given *by him in the 
"Journal of the Franklin Institute" for March of that year. 
If the dotted rectangle ABFG, Fig. 3, shows the original 
outline of the middle section of a nut before punching, he 
found that the final outline of the same section would be 
represented by the full lines. The. top and bottom faces 
were depressed by the punching, as shown ; the upper width 
AB remained about the same, but the lower, GF, was in- 
creased to CD. Although the depth of the nut, AC, was 1.75 
inches, the length of the core punched out was only 1.063 
inches. The density of this core was then examined and 
found to be the same as that of the original nut. Hence a 
portion of the core equal in length to 1.75 — 1.063=0.687 
inch was forced, or flowed, back into the body of the nut. 
Subsequent experiments showed that this flow did not take 
place at the immediate upper surface AB, nor very much 
in the low^er half of the nut, but that it was chiefly confined 
to a zone equal in depth to about half that of the nut, the 
upper surface of which lies a very short distance below the 
upper face of the nut. The location of this zone is shown by 
the lines HK and MA' in Fig. 3. 

Tresca's experiments on punching showed essentially the 
same result. 

Art. 134. — Tresca's Hypotheses. 

The central cylinder FGKH, Fig. i of Art. 133 was called 
by Tresca the "primitive central cylinder." As the metal 
flows, this cylinder will be drawn out into the volume of 
revolution, whose axis is that of the orifice and whose 



812 THE FLOW OF SOLIDS. [Ch. XVIII. 

meridian section is FGkKIIh, Fig. 2, the diameter FG being 
gradually decreased. 

It was found by experim.ent that if the original mass AC, 
Fig. I, was composed of horizontal layers of uniform 'thick- 
ness, the reduced mass in Fig. 2 was also composed of the 
same number of layers of uniform thickness, except in the 
immediate vicinity of the central cylinder. 

Tresca then assumed these three hypotheses: 
1°. — The density of the material remains the same ivhether 
in the cylinder or in the jet; in other words, the volume of the 
material in the jet and in the cylinder remains constant. 

Let R = radius of the cylinder; 
R^ = radius of the orifice ; 
y = variable length of the jet (i.e., hH)\ 
D = original depth of material {BC =AD, Fig. i) 

in the cylinder; 
d = variable depth of material (BC =AD, Fig. 2) 

in the cylinder; 

then by the hypothesis just stated 

R'd=^R'D-R^'y. ...... (i) 

2°. — The rate of compression along any and all lines paral- 
lel to the axis of the primitive central cylinder, and taken outside 
of that limit, is constant. 

If, then, the material. lying outside of the central cylinder 
be divided into horizontal layers of equal thickness, a very 
small decrease in the variable depth equal to d (a) will cause 
the same amount of material to move or flow from each of 
tliese layers into the space originally occupied by the central 
cylinder, thus causing a portion of the material previously 
resting over the orifice to flow through the latter. If d{d) 
vz the indefinitely smah change of depth, and dR^ the in- 
definitely smah change in the radius of the cylindrical por- 



Art. 135.] . THE I/ARIABLH MERIDIAN SECTION. 813 

tion resting over the orifice, tlien the equality of volumes 
expressing this hypothesis is the following: 

K{R'-R,').d{d)^27LR^d.dR^, 

or 

d{d) 2R^dR^ 



d R'-R' 



(2) 



3°. — The rate of decrease of the radius of the primitive cen- 
tral cylinder is constant throughout its length at any given in- 
stant during floiu. 

Let r be any radius less than R^, then if the latter is de- 
creased by the very small amount dR^, the former will be 
shortened by the amount dr; and by the last hypothesis 
there must result 

R^ r ^^' 

This is a perfectly general equation, in which r may or 
may not be the variable value of the radius of that portion 
of the primitive central cylinder remaining above the orifice 
at any instant during flow. 

These are the three hypotheses on which Tresca based 
his theory of the flow of solids. It is thus seen to be put 
upon a purely geometrical basis, entirely independent of the 
elastic or other properties of the material. 

Art. 135. — The Variable Meridian Section of the Primitive 
Central Cylinder. 

The meridian curve hali, or hbK, Fig. 2 of Art. 133, 
may now easily be determined. 

Eq. (i) of Art. 134 may take the first of the following 



8i4 THE FLOIV OF SOLIDS. [Ch. XVIII. 

forms, while its differential, considering d and y variable, 
may take the second: 

R 2 

d(d)==-^dy. 



Dividing the second by the first, 

d{d) dy 2RAR, 



R' ^ R'-R, 

y-R?^ 



The last member of this equation, is simply eq. (2) of 
Art. 134; and if the value of dR^, in eq. (3) of the same 
article, be inserted in the third member of this equation, 
there will result 

2R^^ dr dy 



R'-R'- r R 



y-^.D 



Integrating between the limits of r and R^, and remem- 
bering that r will be restricted to the representation of the 
radius of that portion of the primitive central cylinder 
which remains, at any instant, over the orifice, by taking 
y = o for r = R^, 

r , ly-R:^^ 



log — ^log 



R' 



*'log'' indicates a Napierian logarithm, 



Art. 136.] POSITIONS IN THE JET OF HORIZONTAL SECTIONS. 815 

Passing from logarithms to the quantities themselves, 
and reducing, 

-C-fe)"*] <■> 

This is the desired equation of the line, in which r is 
measured normal to the axis of the cylinder or jet, while y 
is measured along that axis from the extremity of the jet. 
When the material is wholly expelled, 

y = —D, and r = o. 

Eq. (2) is applicable to the jet only. For the line hF or 
Gk, resort will be had to the equation 

d(d) _ 2R,' dr 
d ~R'-R^' r' 

Again integrating between the limits d and D, or r and 
R^, and reducing, 

d\ '^^' 



This value of r is the radius of that portion of the primi- 
tive central cylinder which remains over the orifice when D 
is reduced to d. 



Art. 136. — Positions in the Jet of Horizontal Sections of the 
Primitive Central Cylinder. 

That portion of the primitive central cylinder below ab, 
in Fig. I of Art. 13 3 will be changed to ahKH in Fig. 2 of 
the same article. 



8i6 THE FLOIV OF SOLIDS. [Ch. XVIII. 

If, in the latter Fig., y is the distance from HK to ab, 
measured along the axis, then the volume of HKab will 
have the value 



. rv 

J 



If d' is the distance ciF^bG, in Fig. i, the equality of 
volumes will give 



r r'dy^R^'{D-d'). 
Eq. (i) of Art. 125 gives 



R^-Rx^ 






R2 






If A/" is the number of horizontal layers required to com- 
pose the total thickness D, and n the number in the depth d', 



Hence 



?U"&) J' 



y'-^A ^-{m) \d (2) 



Art. 137.] FINAL RADIUS OF HORIZONTAL SECTION. 817 

Tresca computed values of y' for some of his experiments 
and compared the results with actual measurements. The 
agreement, though not exact, was very satisfactory. Within 
limits not extreme, the longer the jet the more satisfactory 
was the agreement. 

Art. 137,. — Final Radius of a Horizontal Section of the Primitive 
Central Cylinder. 

Let it be required to determine what radius the section 
situated at the distance d^ from the upper surface of the 
primitive central cylinder will possess in the jet. 

It will only be necessary to put for y in eq. (i) of Art. 
135 the value of y' taken from eq. (i) of Art. 136. This 
operation gives 



Hence 



^/\ .R^ 



r'=RAD) (i) 



If R^ is small, as compared with R, there will result ap- 
proximately 

/d'\y^ 



Art. 138. — Path of Any Molecule. 

The hypotheses on which the theory of flow is based 
enable the hypothetical path of any molecule to be easily 
established. 



8i8 THE FLOIV OF SOLIDS. [Ch. XVIII. 

In consequence of the nature of the motion there will be 
three portions of the path, each of which will be represented 
by its characteristic equation, as follows: 

First, let the molecule lie outside of the primitive central 
cylinder. 

Let R' and H be the original co-ordinates of the mole- 
cule considered, measured normal to and along the axis of 
the cylinder, respectively, from the centre of the orifice HK 
(Fig. I, Art. 133) as an origin, while r and h are the variable 
co-ordinates. 

The first hypothesis, by which the density remains con- 
stant, then gives the following equation: 

7t{R'-R'')H = -{R''-r'')K 
or 

hR'~hr' = {R'-R")H (i) 

This is the equation to the path of the molecule, in 
which r must always exceed R^. 

As this equation is of the third degree, the curve cannot 
be one of the conic sections. ■ 

Second, let the m^olecide move in the space originally occu- 
pied by the central cylinder. 

While h and r now vary, the volume 7:r^(D~h) must 
remain constant. When r^R^^ let h=h^. Hence 

r\D-h)=R^'(P-\), ..... (2) 

But if h=\ and r = R^ in eq. (i), 

Placing this value in eq. (2). 

r\D-h)=R,'[D-H^^-^^,y . . . (3) 



Art. 138.] PATH OF ANY MOLECULE, 81Q 

Third, let the molecule move in the jet. 

After the molecule passes the orifice, its path will evi- 
dently be a straight line parallel to the axis of the jet. Its 
distance r^ from that axis will be found by putting h=o m 
eq. (3). Hence 



APPENDIX 1. 

ELEMENTS OF THEORY OF ELASTICITY IN 
AMORPHOUS SOLID BODIES. 



CHAPTER I. 

GENERAL EQUATIONS. 

Art. I. — Expressions for Tangential and Direct Stresses in Terms 
of the Rates of Strains at Any Point of a Homogeneous Body. 

Let any portion of material perfectly homogeneous be 
subjected to any state of stress whatever. At any point as 
Oy Fig. I, let there be assumed any three rectangular co- 
ordinate planes; then consider any small rectangular par- 
allelopiped whose faces are parallel to those planes. Finally 
let the stresses on the three faces nearest the origin be re- 
solved into components normal and parallel to their planes 
of action, whose directions are parallel to the co-ordinate 
axis. 

The intensities of these tangential and normal compo- 
nents will be represented in the usual manner, i.e., p.,3, signi- 
fies a tangential intensity on a plane normal to the axis of 
X (plane ZY), whose direction is parallel to the axis of 
y, while pxx signifies the intensity of a normal stress on 

820 



Art. I.] 



TANGENTIAL AND DIRECT STRESSES. 



821 



a plane normal to the axis of X (plane ZY) and in the 
direction of the axis of X. Two unlike subscripts, there- 
fore, indicate a tangential stress, while two of the same kind 
signify a normal stress. 




Fig. I. 

From eq. (3), Art. 2, and eq. (7), Art. 5, there is at 
once deduced 



5 = 



2(i+r) 



^=Gcl>. 



(i) 



Now when the material is subjected to stress the lines 
bounding the faces of the parallelepiped will no longer be 
at right angles to each other. It has already been shown 
in Art. 2 that the angular changes of the lines from right 
angles are the characteristic shearing strains, which, multi- 
plied by Gs give the shearing intensities. 

Let ^^ be the change of angle of the boundary lines 
parallel to X and Y. 

Let (^2 ^^ "t^^ change of angle of the boundary lines 
parallel to Y and Z. 

Let ^3, be the change of angle of the boundary line 
parallel to Z and X. 



822 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

Eq. (i) will then give ^he following three equations: 

E ^ 

^^y^TiTTr)'!'^'^ (2) 

E ^ 

^^^==7(rT7)^2^ ...... (3) 



E ^ 

^-=7(7T7)^3 (4) 



In Fig. I let the rectangle agfh represent the right pro- 
jection of the indefinitely small parallelepiped dx dy dz. If 
u, V, and w are the unit strains parallel to the axes of x, y, 
and z of the original point h, the rates of variation of strain 

-;-, -r, -T-i etc., may be considered constant throughout 

dx dy dz 

this parallelopiped ; consequently the rectangular faces will 
change to oblique parallelograms. The oblique parallelo- 
gram dhck, whose diagonals may or may not coincide with 
those of agjh, therefore, may represent the strained con- 
dition of the latter figure. 

Then, by Art. 2, the difference petween dhc and the right 
angle at h will represent the strain ^^. But, from Fig. i, ^^ 
has the following value: 

cl)^=dhe-\-bhc. ....... (5) 

But the limiting values of the angles in the second mem- 
ber are coincident with their tangents ; hence 

de be .^. 



Art. I.] STRESSES IN TERMS OF STRy^INS. 823 

But, again, de is the distortion parallel to OX found by 
moving parallel to OY only; hence it is a partial differential 
of w, or it has the value 

'^=^'^y (7) 

In precisely the same manner be is the partial differential 
of V in respect to x, or 

bc = ^-dx. 
dx 



By the aid of these considerations, eq. (6) takes the form 

du dv 

'i'^-Ty+d^-- • • • ■ ■ (8) 

If A^y be changed to YZ, and then to ZX, there may be 
at once written by the aid of eq. (8) 

dv dw . 

'^-=dz"'dy\ (9) 

dw du . . 

^'=5;f+5?- • (^°> 

Eqs. (2), (3), and (4) now take the following form: 

^(dii dv\ . . 

^/dv dw\ , ^ 

^/dw du\ . . 

^-=^5^+^; <'3) 



824 ELASTICITY IN AMORRHOUS SOLID BODIES. [Ch. I. 

The direct stresses are next to be given in terms of the 
displacements u, v, and w. Again, let the rectangular par- 
allelopiped dx dy dz be considered. Eq. (i), on page 3, 
shows that the strain per unit of length is found by dividing 
the intensity of stress by the coefficient of elasticity, if a sin- 
gle stress only exists. But in the present instance, any state 
of stress whatever is supposed. Consequently the strain 
caused by p^^, for example, acting alone must be combined 
with the lateral strains induced by pyy and p^.. Denoting 
the actual rates of strain along the axes of X, Y, and Z by 
/j, l^, and Z3, therefore, the following equations may be at once 
written by the aid of the principles given on pages 9 and 10 : 

^'=k+(Pyy + Pj^'^ .... (14) 

^=h+(p..+PJ^'^ .... (15) 



Eliminating between these three equations, 

?»«=rf,[u^(^.+4+^3)]; . • (17) 



^w„ 



yy i+^l 2 



r^z I +rL ^ I — 2f^ ^ ^ J 



But if ti, V, and w are the actual strains at the point where 
these stresses exist, the rates of strain l„ l^, and l^ will evi- 



Art. I.] STRESSES IN TERMS OF STRAINS. 825 

dently be equal to j^,j-^ and T7, respectively. The volume 
of the parallelopiped will be changed by those strains to 

dx{i+l^)dy(i+l^)dz{i+l^) =dx dy dz{i +1^ + 1^^-!^) 

if powers of l^, l^, and l^ above the first be omitted. The 
quantity (l^-hl^ + h) is, then, tJie rate of variation of volume, 
or tlie amount of variation of volume for a cubic unit. If 
there be put 

^ dti dv dzv ^ ^ E 

= :t-, + -j-+-j-, and G = 



dx ' dy dz' 2(1 +r)' 

eqs. (17), (18), and (19) wih take the forms 

2Gr . ^du . .. 

P^^-T^r^+'^dx' • • • • (2°) 

2Gr „ ^dv 

2Gr dw 
p ,= d^2G-T- (22) 

The form in which eqs. (14), (15), and (16) are written 
shows that if p^^, pyy, or p^^ is positive, the stress is tension, 
and compression if it is negative. Consequently a positive 
value for any of the intensities in eqs. (20), (21), or (22) will 
indicate a tensile stress, while a negative value will show 
the stress to be compressive. 

The eqs. (14) to (19), together with the elimination in- 
volved, also show that the coefficients of elasticitv for ten- 
sion and compression have been taken equal to each other, 
and that the ratio r is the same for tensile and compressive 
strains. 



826 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

Further, in eqs. (ii), (12), and (13), it has been assumed 
that G is the same for all planes. 

Hence eqs. (11,) (12), (13), (20), (21), and (22) apply 
only to bodies perfectly homogeneous in all directions. 

It is to be observed that the co-ordinate axes have been 
taken perfectly arbitrarily. 

Art. 2. — General Equations of Internal Motion and Equilibrium. 

In establishing the general equations of motion and equi- 
librium, the principles of dynamics and statics are to be 
applied to the forces which act upon the parallelopiped repre- 
sented in Fig. I , the edges of which are dx, dy, and dz. The 
notation to be used for the intensities of the stresses acting 
on the different faces will be the same as that used in the 
preceding article. 

Let the stresses which act on the faces nearest the origin 
be considered negative, while those which act on the other 
three faces are taken as positive. 

The stresses which act in the direction of the axis of X 
are the following: 

On the face normal to X, nearest to 0, — p^^ dy dz ; 

" ''isiTthestiTomO,(p^^ + -~^dxjdydz; 
* dy dx nesLvest to 0, —p^^dydx; 



it (i 



farthest from 0, (p.^ + ~T^^^ ]dy dx ; 



dz dx nearest to 0, —p y^ dz dx ; 

** ** farthest from 0, ipy^ + -j^dyjdzdx. 



Art. 2.] EQU/tTIONS IN RECTANGULAR CO-ORDINATES. 



827 



clz 



dx 



dy 



The differential coefficients of the intensities are the rates 
of variation of those intensities for each unit of the variable, 
which, multipHed by the 
differentials of the varia- 
bles, give the amounts of 
variation for the different dz 

edges of the par allelopiped . |d^ 

Let Xq be the external dx 

force acting in the direc- 
tion of X on a unit of vol- 
ume at the point consid- 
ered ; then X^dxciy dz will 
be the amount of external 
force acting on the paral- Fig. i. 

lelopiped. 

These constitute all the forces acting on the parallelo- 

piped in the direction of the axis of X, and their sum, if un- 

d'^u 
balanced, must be equal to m-rr^dx dy dz ; in which m is the 

mass or inertia of a unit of volume, and dt the differential 
of the time. Forming such an equation, therefore, and drop- 
ping the common factor dx dy dz, there will result 



dx ^ dy ^ dz ^^'>- "^dt^- 



(i) 



Changing x to y, y to z, and z to x, eq, (i) will become 
+ -zJ7f+-^nr+yo = nt:Tr^' ... (2) 



dx ^ dy ^ ^- ^^' "^ 



d'. 



Again, in eq. (i), changing x to z, z to y, and y to x, 



dx ^ dy '^ 



^7 _ ^ 
dz ^"^'-"^dt'' 



(3) 



828 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

The line of action of the resultant of all the forces which 
act on the indefinitely small parallelopiped, at its limit, 
passes through its centre of gravity, consequently it is sub- 
jected to the action of no unbalanced moment. The parallelo- 
piped, therefore, can have no rotation about an axis passing 
through its centre of gravity, whether it be in motion or 
equilibrium. Hence, let an axis passing through its centre 
of gravity and parallel to the axis of X, be considered. The 
only stresses, which, from their direction can possibly have 
moments about that axis, are those with the subscripts (yz), 
{zy), {yy), or {zz). But those with the last two subscripts 
act directly through the centre of the parallelopiped, conse- 

dp 
quently their moments are zero. The stresses ~r^^dy .dx dz 

dpz 
and — T^ dz . dx dy are two of six forces whose resultant is 

directly opposed to the resultant of those three forces which 
represent the increase of the intensities of the normal, or 
direct, stresses on three of the faces of the parallelopiped; 
these, therefore, have no moments about the assumed axis. 
The only stresses remaining are those whose intensities are 
pzy and pys. The resultant moment, which must be equal 
to zero, then, has the following value: 

py^dx dz.dy + pzydx dy .dz = o\ ... (4) 

' .*. Pyz=-Pzy (5) 

Hence the two intensities are equal to each other. 

The negative sign in eq. (5) simply indicates that their 
moments have opposite signs or directions; consequently, 
that the shears themselves, on adjacent faces, act toward 
or from the edge between those faces. In eqs. (i), (2), and 
(3), the tangential stresses, or shears, are all to be affected 



Art. 2.] EQUATIONS IN RECTANGULAR CO-ORDINATES. 829 

by the same sign, since direct, or normal, stresses only can 
have different signs. 

The eq. (5) is perfectly general, hence there may be 
written : 

P.y=Py.^ and p,,=p,, (6) 

Adopting the notation of Lame, there may be written: 

P..=^\^ Pyy-^\^ P..-^\\ 
Pzy^^v Pxz = T^2^ Pxy-^z\ 

by which eqs. (i), (2), and (3) take the following forms: 
dN, dT, dT, ,, dhi 

dT, dN, , dT, ^^ 

dT, dT, dN, ^ 
'd^ + ^^-df+^^-'^ 

The equations (11), (12), (13), (20), (21), and (22) of the 
preceding article are really kinematical in nature ; in order 
that the principles of dynamics may hold, they must satisfy 
eqs. (7), (8), and (9). As the latter stand, by themselves, 
they are applicable to rigid bodies as well as elastic ones; 
but when the values of A^ and T, in terms of the strains u, v, 
and w, have been inserted, they are restricted, in their use, 
to elastic bodies only. With those values so inserted, they 
form the equations on which are based the mathematical 
theory of sound and light vibrations, as well as those of 
elastic rods, membranes, etc. In general, they are the equa- 
tions of motion which the different parts of the body can 



' dt' ' * 


. . (7) 


d'v 
' df' ' 


. . (8) 


d'w 


. . (9) 



830 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

have in reference to each other, in consequence of the elastic 

nature of the material of which the body is composed. 

If all parts of the body are in equilibrium under the 

action of the internal stresses, the rates of variation of the 

d'^u d^v . d^w -11 1 i 

strams -77^, -7tf» and -7-^, will each be equal to zero. 

Hence, eqs. (7), (8), and (9) will take the forms 

dN, dT, dT, ^ 

-57 + ^ + ^+^0 = 0;. . . . (10) 

dTo dN^ dT. ^_ , ^ 

itt+ify+-dr+^'-°'- • ' ' (") 

dT, dT, , dN, ^ , ^ 

-dt + ^ + lk+^'-° (-) 

These are the general equations of equilibrium. As they 
stand, they apply to a rigid body. For an elastic body, the 
values of N and T from the preceding article, in terms of the 
strains n, v, and w, must satisfy these equations. 

The eqs. (10), (11), and (12) express the three conditions 
of equilibrium that the sums of the forces acting on the 
small parallelopiped, taken in three rectangular co-ordinate 
directions, must each be equal to zero. The other three con- 
ditions, indicating that the three component moments about 
the same co-ordinate axes must each be equal to zero, are 
fulfilled by eqs. (5) and (6). The latter conditions really 
eliminate three of the nine unknown stresses. The remaining 
six consequently appear in both the equations of motion 
and equilibrium. 

The equations (7) to (12), inclusive, belong to the interior 
of the body. At the exterior surface, only a portion of the 
small parallelopiped will exist, and that portion will be a 



Art. 2.] EQUATIONS JN RECTANGULAR CO-ORDINATES. 831 

tetrahedron, the base of which forms a part of the exterior 
surface of the body, and is acted upon by external forces. 

Let — be the area of the base of this tetrahedron, and let 
2 . 

p, q, and r be the angles which a normal to it forms with 

the three axes of X, Y, Z, respectively. Then will 

da cos p =dy dz, da cos q=dz dx, and da cos r -=dx dy. 

Let P be the known intensity of the external force acting 
on da, and let tt, /, and p be the angles which its direction 
makes with the co-ordinate axes. Then there will result : 

Xq=P da. cos 7z, Yq=P da.cos Xy and Zq=P da. cos p. 

The origin is now supposed to be so taken that the apex of 
the tetrahedron is located between it and the base; hence 
that part of the parallelopiped in which acted the stresses 
involving the derivatives, or differential coefficients, is 
wanting ; consequently those stresses are also wanting. 

The sums of the forces, then, which act on the tetra- 
hedron, in the co-ordinate directions, are the following: 

— (A\ dy dz -{- T.^ dz dx + T^ dy dx) + Pda cos 7t=o\ 

— (Tg dz dy + N, dz dx 4- T^ dy dx) -h Pda cos ^ = o ; 

— ij^ dz dy + T^ dz dx + A^g dy dx) + Pda cos ^ = o. 

Substituting from above, 

A^j cos ^+ Tg cos g + 72 cos r = P cos tt; . . (13) 

T^cosp\N^cosq^T^co?>r=P cos i\ . . (14) 

r^ cos ^ + r^ cos (7 + A^g cos r = P cos ^. , . (15) 

These equations must always be satisfied at the exterior 
surface of the body; and since the external forces must 
always be known, in order that a problem may be determi- 
nate, they will serve to determine constants which arise 



832 



ELASTICITY IN AMORPHOUS SOLID BODIES. 



[Ch. I. 



from the integration of the general equations of motion and 
equihbrium. 

Art. 3. — Equations of Motion and Equilibrium in Semi-polar 

Co-ordinates. 

For many purposes it is convenient to have the condi- 
tions of motion and equilibrium expressed in either semi- 
polar or polar co-ordinates ; the first form of such expression 
will be given in this article. 

The general analytical method of transformation of co- 
ordinates may be applied to the equations of the preceding 
article, but the direct treatment of an indefinitely small 
portion of the material, limited by co-ordinate surfaces, pos- 
sesses many advantages. In Fig. i are shown both the 





#^ 






K 


X ^ 




dx 
h 


>--^^ 

^iv 


y 


e 


; 1 \ 

1 1 




Y 


s 

N 


1 


> 


'^ 



nM- 



Fig. I. 



Y 



small portion of material and the co-ordinates, semi-polar 
as well as rectangular. The angle made by a plane normal 
to ZY, and containing OX, with the plane XY is repre- 
sented by (f) ; the distance of any point from OX, measured 
parallel to ZY, is called r; the third co-ordinate, normal to 



Art. 3.] EQUATIONS IN SEMI-POLAR CO-ORDINATES, 833 

r and ^, is the co-ordinate x, as before. It is important to 
observe that the co-ordinates x, r, and (f>, at any point, are 
rectangular. 

The indefinitely small portion of material to be con- 
sidered will, as shown in Fig. i , be limited by the edges dx, dr^ 
and r d(j). The faces dx dr are inclined to each other at the 
angle d(f). 

The intensities of the normal stresses in the directions of 
X and r will be indicated by A^^ and R, respectively. The 
remainder of the notation will be of the same general char- 
acter as that in the preceding article; i.e., T^^ will represent 
a shear on the face dr .r dcj) in the direction of r, while N^ is 
a normal stress, in the direction of ^, on the face dx dr. 

The strains or displacements, in the directions of x, r, and 
(j), will be represented by ti, p, and w ; consequently the 
unbalanced forces in those directions, per imit of mass, 
will be 

d^u d^p ^ d^w , ^ 

"^w^ "^w^ ^^^ ""'w (') 

Those forces acting on the faces hf, fe, and he, will be 
considered negative ; those acting on the other faces, posi= 
tive. 

Forces Acting in the Direction of r. 

— R.rdcpdx, and 

-\-Rr dcj) dx+l-^ — dr = r-^dr + R drjdcf) dx. 

— T^rdr dx, and 

+ T^rdrdx + —T^d(l).drdx. 
^T^r-'f d(j) dr, and 

7'X-' 

'\-Txr-fd(l)dr + -j^dx.rd(j)dr, 



834 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

On the face dr dx, nearest to ZOX, there acts the normal 
stress ( N^^dr dx + -j-^d^ .drdx\=N'\ and A^' has a com- 
ponent acting parallel to the face fe and toward OX, equal to 
N' sin {d(f)) =N'^—^=N'd(j). But the second term of this 

product will hold (<i^)^ hence it will disappear, at the limit, 
in the first derivative of N'd(j) .'. N'dcp^N^dcj) dr dx. 
Since this force must be taken as acting toward OX, it 
acts with the normal forces on hf, and, consequently, must 
be given the negative sign. 

If Rq is the external force acting on a unit of volume, 
another force (external) acting along r will he R^.r d(j) dr dx. 

The sum of all these forces will be equal to 

m .rd(j) dr dx . -j-^. 

Forces Acting in the Direction of j). 

— N^dr dx, and 

dN 
+ N^dr dx + \t^ d9 • d^ dx, 

— Tr^-rd(j) dx, and 

-^Tr^.rd^dx+ (^^^^^dr = r^dr + Tr^di\d4> dx, 

— Tx^.r dcf) dr, and 

+ Tx^.rd(f)dr + —-T^dx.rd(f)dr, 

As in the case of A/",^, in connection with the forces along 
r, so the force T^j. dr dx has a component along ^ (normal 
to fe) equal to T^rdrdx. sin {d<j)) =T ^rd<j) dr dx. It will 
have a positive sign, because it acts from OX. 

The external force is (P^.r d(f) dr dx. 



^^' 3-] EQUATIONS IN SEMI-POLAR CO-ORDINATES. 835 



Forces Acting in the Direction of x, 

-N^.r d^ dr, and 

dN 
f A\r dcj)dr+ -f^dx . r d^ dr. 

^Trx'dx r dcj), and 

4- Trx .dxrdcjy + l — , ^^ dr = r~-~dr + Trx drjdx d<}). 

— T^xdx dr, and 

+ T^^dx dr + it^ dcj) . dx dr. 

The external force is Xo.r dcj) dx dr. 

Putting each of these three sums equal to the proper 
rates of variation of momentum, and dropping the common 
factor r d(j) dx dr: 

dx ^ dr + rd<p^ r +^<> ^dP ' * ^"■' 

dT„dRdT,R--N^ d^p_ 



dT,^ dTr4, dN^^ Tr^ + Tr^ _,«^ 
dx '^ dr ^ rd4,- r + '""-^^df 



(4) 



These are the general equations of motion (vibration) in 
terms of semi-polar co-ordinates ; if the second members kre 
made equal to zero, they become equations of equilibrium. 
Eqs. (2), (3), and (4), are not dependent upon the nature of 
the body. 

Since x, r, and ^ are rectangular, it at once follows that 

^ rx — -^ xrj J- r4> = ^ 4>r^ 2-M i x4>^ J- <i>X' • * (5/ 



836 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

In order that eqs. (2), (3), and (4) may be restricted to 
elastic bodies, it is necessary to express the six intensities 
of stresses involved, in terms of the rates of variation of the 
strains in the rectangular co-ordinate directions of x, r, and 
(j). Since these co-ordinates are rectangular, the eqs. (11), 
(12), (13), (20), (21), and (22) of Article i, may be made 
applicable to the present case by some very simple changes 
dependent upon the nature of semi-polar co-ordinates. 

For the present purpose the strains in the co-ordinate 
directions of x, y, and z will be represented by ti\ v', and 
w\ Since the axis of x remains the same in the two systems, 
evidently 

du^ du 
dx dx' 

From Fig. i it is clear that the axis of y corresponds 
exactly to the co-ordinate direction r; hence 

dv^ dp 
dy ~dr' 

From the same Fig. it is seen that the axis of z corre- 
sponds to ^, or r(j). But the total differential, dw\ must be 
considered as made up of two parts ; consequently the rate 

dvu^ 
of variation -j- will consist of two parts also. If there is no 

distortion in the direction of r, or if the distance of a mole- 
cule from the origin remains the same, one part will be 

div dw 
-TT—TT =~-n' If, however, a unit's length of material be re- 
d{r(l)) rdcf) 

moved from the distance r to r + ^ from the centre 0, Fig. i, 

while (j) remains constant, its length will be changed from 

I to (i-f-l, in which p may be implicitly positive or 



Art. 3. J EQUATIONS IN SEMI-POLAR CO-ORDINATES, 
negative. Consequently there will result 



837 



dw' 
dz 



dw p 
rdcbr' 



For the reason already given, there follow 

dii^ du dv^ dp 

i—=~r~ and T~^ = -r-. 
dy dr ax ax 

In Fig. 2 let dc be the side of a distorted small portion 
of the material, the original position ^ 
of which was d'e. Od is the distance 
r from the origin, ad=dr and ac = 
dw, while dd' = w. The angular 

change in position of dc is —, = -,- ; 

^ ah w , 
but an amount equal to —3 = - is due to the movement of 

r, and is not a movement of dc relatively to the material 
immediately adjacent to d. 
Hence 

dp 




Fig. 2. 



dw' _dw w dv' 

d^^d^~^' ^^^^ di 



rd(j)' 



There only remain the following two, which may be at 
once written 

dw' dw 
dx dx 



. du' du 
and -r=—Ti, 
dz rd(j> 



The rate of variation of volume takes the following form 
in terms of the new co-ordinates: 



~^^ dy dz ~dx^dr'^rd4~^r' 



dx 



(6) 



838 ELASTICITY IN AMORPHOUS SOLID BODIES, [Ch. I. 

Accenting. the intensities which belong to the rectan- 
gular system x, y, z, the eqs. (11), (12), (13), (20), (21), and 
(0.2^. of Art. I, take the following form: 



», -'V,-^,«+.^£: (rt 



^ 1-2V dr* ^ ^ 



2Gr - r^( dw p\ 



(9) 



^-^.'=<i'43^ <■»> 

'■..=n'=<^+.T-r)^ <■■> 

n-'-.'=K£+^*)- • • • • • ■("> 

If these values are introduced in eqs. (2), (3), and (4), 
those equations will be restricted in application to bodies 
of homogeneous elasticity only. 

The notation t is used to indicate that the r involved is 
the ratio of lateral to direct strain, and that it has no rela- 
tion w^hatever to the co-ordinate r. 

The limiting equations of condition, (13), (14), and (15) 
of Art. 2, remain the same, except for the changes of nota- 
tion, shown in eqs. (7) to (12), for the intensities N and T. 



Art. 4,J 



EQUATIONS IN POLAR CO-ORDINATES. 



839 



Art. 4. — Equations of Motion and Equilibrium in Polar 
Co-ordinates. 

The relation, in space, existing between the polar and 
rectangular systems of co-ordinates is shown in Fig. i . The 
angle is measured in the plane ZY and from that oi XY; 




Fig. I. 

while (p is measured normal to ZY in a plane which contains 
OX. The analytical relation existing between the two sys- 
tems is, then, the following: 

.x = rsin0, y =r cos ([' cos, 6, and z=--r cos (p sin 6. 

The indefinitely small portion of material to be considered 
IS ah e d. It is limited by the co-ordinate planes located by 
(j) and 0, and concentric spherical surfaces with radii r and 
r -h dr. The directions r, (p, and (p, at any point, are rectangu- 
lar ; hence the sums of the forces acting on the small portion 
of the material, taken in these directions, must be found and 
put equal to 



m 



'dp ' 



m 



dt' 



and 



m 



dt' 



840 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

in which expressions, p, 7;, and co represent the strains in the 
direction of r, ^, and ^ respectively. 

Those forces which act on the faces ah, bd, and cd will be 
considered negative, and those which act on the other faces 
positive. 

The notation will remain the same as in the preceding 
articles, except that the three normal stresses will be indi- 
cated by Nr, N^, and N^. 

Forces Acting Along r. 

— Nr.r d(ff r cos 4f dcf), 

i-Nr.r^ cos (l^dil^dcj) 

/d(N r^) dN \ 

"^ ( dr '^ ^ ^'^^^+ 2rNrdrj cos (p dip d<t>, 

-T^r-rd([fdr. 

+ T^r .rdilfdri- jt^ d<j) :r d (p dr, 

— r^.r cos (p d(f) dr, 
+ T^.r cos (p dcj) dr 

_^/ d{T^rCOS iP) ^^ ^ ^^^ cp^'diP-T^,sm^dcp]rd<pdr, 

— N<i> .r dip dr . sin aOc = — A^^ .r dip dr . cos ip d<j), on face ce, 

— N4,.r cos ip dcp dr. sin aOb = —N^.r cos ip d(p dr.dip, 

on face be. 

Forces Acting Along j), 

— Trd^.r cos ip d^ r d (I). 

■{■Tr4>-r'^ COS ip dcp dip 

+ ( '^^y^ ^r= r''^dr + 2r Tr^dr) cos <p diP dcj,. 



Art. 4.J EQUATIONS IN POLAR CO-ORDINATES. 841 

— N<j> .r d(p dr. 

+ N4>.rdiljdr + -j^d^rd([fdr. 

— T^^.r cos (p dcj) dr, 
-{jT^^cos (p.r dcf) dr 

\ d'' — ^^^^^^ 4^~^7d4'-T^^s\n (pdipjrd^dr. 

+ T^rf' d 4^ dr. cos i[f dcj), on face c^. 

— T^^ r dip drl sin akc = — y— ) = — T^^ r dip dr. sin if' d(f)y 

on face ce. 

The lines ak and ck are drawn normal to Oc and Oa, 

Forces Acting Along (p. 

— Tr^.r cos ip d(j).r dip. 
■\-Tr4,r^ cos ip dcp di[} 

'dr+2rTr^drj cos (pdipdcp. 

— T^^.rd.i/' dr. 

+ T^4. r dip dr + -j-^dcp .r dip dr. 

— N^.r cos ip dcj) dr. 

■\-N^.r cos ip dcp dr ' 

./d(N^cosiP),^ dN^ . \ 

+ I ^. d ip = cos ip-T-fdip — N^sm ip dip j r dcp dr. 

\-T^.r cos ip dcp dr .dip, on isice be. 

f A^,^ .r dip dr. sin akc = + N^f, r dip dr. sin ip dcp, on face ce. 



-^^-^^''^' -dr 



842 ELASTICITY IN AMORPHOUS SOLID BODIES. [Ch. I. 

The volume of the indefinitely small portion of the 
material is (omitting second powers of indefinitely small 
quantities) 

r cos (Jf d(j).r d(fi.dr = AV , 

and its mass is m multiplied by this small volume. The 
latter may be made a common factor in each of the three 
sums to be taken. 

The external forces acting in the directions R, (j), and (p 
, will be represented by 

RJV, 0JV, and WJV, 
respectively. 

Taking each of the three sums, already mentioned, and 
dropping the common factor J F, there will result 



d^ dT^r ^ dT^,r ^ 2 Nr - N^ - A^^ - T^, tan </> 
dr r cos (p.d^ rdip r . 



d^p 
'df 



+ Ro=^^^M^> (i) 



dTrj dN^_ dT^^ 

dr r cos (p .dcp r dip 

,^tan<^-r^^tan(/; ^^^ „.. 

dt 



2Tr^ + T4>r-T^^tEin(l>-T^^tan(p d^-q 

+ + <^o = ^«Tri^; (2; 



dT,, ^ dT,^ ^ dN^ 



dr r cos (l^d(\) r d(p 

2 Tr^, + T,, - N^ tan y^ + A^^ tan ^^ _ d^ 

Since r, -/>, and ^ are rectangular at any point, 



Art. 4. J EQUATIONS IN POLAR CO-ORDINATES, 843 

Hence 

r r ' 

2 Tr^ + T^r - tan yA ( jV^ - A ^^) _ 3 r,^, - tan <^ (.V^ - .V^) 
r r ' 

These relations somewhat simplify the first members of 
eqs. (2) and (3). 

Eqs. (i), (2), and (3) are entirely independent of the 
nature of the material ; also, they apply to the case of equi- 
librium, if the second members are made equal to zero. 

The rectangular rates of strain, at any point, in terms 
of r, (j), and (/^ are next to be found. As in the preceding 
article, the rates of strain in the rectangular directions of 
r, (j), and (p will be indicated by 

dv^ dw' du' dv^ du' 

Sy' 57"' d^' d^' 57' ^^^• 

Remembering the reasoning in connection with the value 

dw^ 
of ~7— , in the preceding article, and attentively considering 

Fig. I, there may at once be written, 

dtt^ doj p 
dx' r d if) f 

In Fig. I, if ac = I and ah = uj, while ak =r cot. (p (ak is 
perpendicular to aO), the difference in length between ac 
and bh will be 

CO CO tan (p 

rcot d) r 

This expression is negative because a decrease in length takes 
place in consequence of a movement in the positive direction 1 
of nl). 



844 



ELASTICITY IN AMORPHOUS SOLID BODIES. 



[Ch. I. 



Again, a consideration of Fig. i, and the reasoning con- 
nected with the equation above, will give 

dw' drj _ p coicHKp 



rcoSil>d(j) r 



Without explanation there may at once be written: 

dv^ _dp 
d^'~dr' 

Fig. I of this, and Fig. 2 of the preceding article, give 

du' doj CO . dv' dp 

dy' dr r 



and 



rd(lf' 



Precisely 



dr r dx' 

These are to be vised in the expression for T^ 
the same figures and method give 

dv^ dp dw' df] T) 

dz' rcosil^dcj) " dy dr r' 

which are to be used in finding T,^;.. 

The expression for -r-j- will be composed of the sum of 

two parts. In Fig. 2, ah is the original position of r d(^>, and 
after the strain t] exists it takes the position ec. Consequently 
ac (equal and parallel to bd and perpen- 
dicular to ak) represents the strain tj, 
while ed represents drj. vSince, also, fc is 
perpendicular to ck, the strains of the kind 
Tj change the right angle fck to the angle 
fee; or the angle eck is equal to 




dw' 

d^=''^- 
df) 



. - ed ca 

dck = -1- + — r 

do ak 



Fig. 



r dil) r cot (^' 
In Fig. 2, the points a, 6, and k are 
identical with the points similarly lettered in Fig. i. The 



Art. 4. J EQUATIONS IN POLAR CO-ORDINATES. 845 

expression for tt ^^^Y be at once written from Fig. i. There 
may, then, finally be written, 

dw^' df] 7) tan ^ . du' _ dco 

dx' ~~rd(j) r dz' r cos ^ d(j)' 

These equations will give the expression for T^^, 
The value of 

du^ dv' dw' 
^^M^d^^'^W 

now takes the following form: 

^ do df] dii) 2p 6; tan <p 

c' = "V- -1 n~L "1 — 1~, + ... (4) 

dr rcos(pd(j) rd(p r r ^^' 

The last two terms are characteristic of the spherical 
co-ordinates. 

The eqs. (20), (21), (22), (11), (12), and (13), of Art. 
I, take the forms 

^-i^/^-4:^ (s) 

2GX ^ ^l df] p ct* tan S\ ^ ^ 

A.,=^^ + ,Gte+^); (7) 

^ \ — 2X \rd>l) rj ^" 

^** = ^(nft. + r cos "i d^ + ^L^j . .... (8) 

( dp df) r)\ 
^ '*^'^\r cos ^d4>^d~r 7)' ^^°> 



846 ELASTICITY OF AMORPHOUS SOLID BODIES. [Ch. I. 

If these values are inserted iii eqs. (i), (2), and (3), the 
resulting equations will be applicable to isotropic material 
only. 

As in the preceding article, V is used to express the ratio 
between direct and lateral strains, and has no relation what- 
ever to the co-ordinate r. 

It is interesting and important to observe that the equa- 
tions of motion and equilibrium for elastic bodies are only 
special cases of equations which are entirely independent of 
the nature of the material, of equations, in fact, which 
express the most general conditions of motion or equilibrium. 



CHAPTER IT. 

THICK, HOLLOW CYLINDERS AND SPHERES, AND 
TORSION. 

Art. 5.— Thick, Hollow Cylinders. 

In Fig. I is represented a section, taken normal to its 
axis, of a circular cylinder whose walls are of the appreciable 
thickness t. Let p and p^^ represent the interior and exterior 
intensities of pressures, respectively. The material will not 
be stressed with uniform intensity throughout the thickness /. 
Yet if that thickness, comparatively 
speaking, is small, the variation will 
also be small; or, in other words, 
the intensity of stress throughout 
the thickness t ma}^ be considered 
constant. This approximate case 
will first be considered. 

The interior intensity p will be 
considered greater than the exterior 
p^, consequently the tendency will 
be toward rupture along a diametral plane. If, at the same 
time, the ends of the cylinder are taken as closed, as will be 
done, a tendency to rupture through the section shown in the 
figure will exist. 

The force tending to produce rupture of the latter kind 
will be 

F^7:(pr^'-p,r,') (i) 

847 




848 THICK, HOLLOIV CYLINDERS. [Ch. II. 

If N^ represents the intensity of stress developed by this 
force, 

If the exterior pressure is zero, and if r' is nearly equal to 

r, + r' 
2 

TV— ^^ = ^ r\ 

^"^^ 2(r,-/) 2t ^^^ 

In this same approximate case, the tendency to split the 
cylinder along a diametral plane, for unit of length, will iDe 

If A/"' is the intensity of stress developed by F\ 

^ =T^. — t — • (4) 

xV is thus seen to be twice as great as N^ when p^ = o. If, 
therefore, the material has the same ultimate resistance in 
both directions the cylinder will fail longitudinally w^hen the 
interior intensity is only half great enough to produce trans- 
verse rupture, the thickness being assumed to he very small and 
the exterior pressure zero. 

N^ and N' are tensile stresses, because the interior pres- 
sure w^as assumed to be large compared with the exterior. If 
the opposite assumption were made, they would be found to 
be compression, while the general forms would remain ex- 
actly the same. 



AjL 5.] THICK, HOLLOIV CYLINDERS. 849 

The preceding formulas are too loosely approximate for 
many cases. The exact treatment requires the use of the 
general equations of equilibrium, and the forms which they 
take m Art. 3 are particularly convenient. As in that article, 
the axis of x will be taken as the axis of the cylinder. 

Since all external pressure is uniform in intensity and 
normal in direction, no shearing stresses will exist in the 
material of the cylinder. This condition is expressed in the 
notation of Art. 3 by putting 

T^x = Tfx = Tf.^ = o. 

Again the cylinder will be considered closed at the ends, 
and the force F, eq. (i), will be assumed to develop a stress 
of uniform intensity throughout the transverse section 
shown in Fig. i. This condition, in fact, is involved in that 
of making all the tangential stresses equal to zero. 

Since this case is that of equilibrium, the equations (2), 
(3), and (4) of Art. 3 take the following form, after neglect- 
ing Xo, Rq, and 0q'. 

■5^ = °^ (5) 



f +5^=0. (« 



-VT4>''° (7) 

These equations are next to be expressed in terms of the 
strains u, p, and w. 

In consequence of the manner of application of the exter- 
nal forces, all movements of indefinitely small portions of 



850 THICK, HOLLOIV CYLINDERS. [Ch. II. 

the material will be along the radii and axis of the cylinder. 
Hence 

ti will be independent of r and (p; 

p ^ ^; 

The rate of change, therefore, of volume will be (eq. ^6) 
of Art. 3) 

du dp p 

dx dr r ^ '' 



* . • 1 1 ^ dd dhi 1 ' r 

As p IS mdependent of x, ~^^ =TT2 i hence if the value of 

N^ be taken from eq. (7) of Art. 3 and put in eq. (5) of this 
article, 

dK\ 2GX dhi ^dhi_ 
dx ~i-2Vdx'^^^"dx'~^' 

d'u 

.*. -J— 2 = and ti---^ax + a. 

But the transverse section in which the origin is located 
may be considered fixedt. Consequently if x-~=q, u=q and 
thus a' =0. The expression for u is then u =ax. 

The ratio u-'rx is the / of eq. (i), on page 3, while the 
p of the same equation is simply N^ of eq. (2), given above. 
Hence 



Art. 5. J THICK, HOLLO JV CYLINDERS. 851 

Again, eq. (8) of Art. 3, in connection with eqs. (8) 
and (6) of this, gives 

2GV /d'p dp _p\^^Jd^ j^Ap._(!\ =0 
i.— 2X\dr'^ r dr r^J ' " \dr^ r dr r'y 

d(^' 
^ d'p ^ dp _p _d"y V, _^ 

' * dr^ r dr r^ dr"^ dr 

dp p 
,\ ~ + -=c, or 
dr r 

r dp + p dr^dypr) ^-cr dr. 

cr"^ ^ cr b , ^ 

.*. pr=— + b, or P = j+-- . . . (10) 

This value of p in eqs. (8) and (9) of Art. 3 will give 
iV(a + c) c b) 

At the interior surface R must be equal to the internal 
pressure, and at tlie exterior surface to the external pressure. 
Or since negative signs indicate compression, 

If r =/.... , R=~p, 
li r=r^ . . . . . R=-p^. 

Either of these equations is the simple result of applying 
eqs. (13), (14), and (15) to the present case, for which 

cos /? = cos r = cos ;r = cos ,0 = o, 
cos q = cos X = I , and P -= — p or — p^. 



852 THICK, HOLLOIV CYLINDERS. [Ch. II. 

Applying eq. (11) to the two surfaces, 

Subtracting (14) from (13), 

r'^ — r^ 
Inserting this value in eq. (13), 

( I — 2t 2' 

The general expressions of R and A^,^^, freed from the 
arbitrary constants of integration, can now be easily written 
by inserting these last two values in eqs. (11) and (12). By 
making the insertions there will result 

The stress A^,^<^ is a tension directed around the cylinder, 
and has been called "hoop tension." Eq. (16) shows that the 
hoop tension will be greatest at the interior of the cylinder. 
An expression for the thickness, t, of the annulus in terms of 
the greatest hoop tension (which will be called h) can easily 
be obtained from eq. (16). 



Art. 6.] TORSION IN EQUILIBRIUM, 853 

If r =r' in that equation, 



h = 



•• / \2p,-p + hJ ' 

.■..-=.H(;?r^)'-!----" 

Eq. (17) will enable the thickness to be so determined 
that the hoop tension shall not exceed any assigned limit h. 
If p^ is so small in comparison with p that it may be neg- 
lected, t v/ill become 

H(?7^)'-l '■« 

If p^ is greater than p, N'<pcf> becomes compression, but 
the equations are in no manner changed. 

The values of the constants b and c may easily be found 
from the two equations immediately preceding eq. (15). 

It is interesting to notice that the rate of change of vol- 
ume, 6, is equal to (a + c) and therefore constant for all 
points. 

Art. 6. — Torsion in Equilibrium, 

The formulas to be deduced in this article are those first 
given by Saint -Venant, and established in substantially the 
same manner. 

It will in all cases, except that of the final result for a 
rectangular cross-section, be convenient to use those equa- 
tions of Art. 3 which are given in terms of semi-polar co- 
ordinates. 



854 



TORSION IN EQUILIBRIUM. 



[Ch. II. 



Let Fig. I represent a cylindrical piece of material, with 
any cross-section, fixed in the plane ZY, and let the origin of 

co-ordinates be taken at 0. Let 
it be twisted also by a couple 

P.ab=Pl, 

the plane of which is parallel to 
ZY. The material will thus be 
subjected to no bending, but to 
pure torsion. 

The axis of the piece is sup- 
posed to be parallel to the axis 
of X as well as the axis of the 
couple. Normal sections of the 
piece, originally parallel to ZOYy 
will not remain plane after tor- 
sion takes place. But the tendency to twist any elementary 
portion of the piece about an axis passing through its centre 
and parallel to the axis of X will be very small compared 
with the tendency to twist it about either the axis of r or <A; 
consequently the first will be neglected. In the notation 
of Art. 3, this condition is equivalent to making T^^ = o. 

As the piece is acted, upon by a couple onb/, all normal 
stresses will be zero. 

Eqs. (7), (8), (9), and (11) of Art. 3 then become 




Fig. 



2GV „ ^du 
^ 1 - 2V ax 



o; 



R=- 



2GX 



N, 



[-2r 
2GX 






dw 
■2t \rd<j> 



r 



■''* ^rd4> ' dr r 



o; 



(1) 
(2) 
(3) 

(4) 



Art. 6.] TORSION IN EQUILIBRIUM. . 855 

After introducing the values of T^x and T^x, from eqs. 
(10) and (12) of Art. 3, in eqs. (2), (3), and (4) of the same 
article, at the same time making the external forces and 
second members of those equations equal to zero, and bear- 
ing in mind the conditions given above, there will result 



dr rd^ r 

/dJu d^p d^w d^u dii ^^p\ _ 

^^\dr^'^dFdx'^rdJdx'^?d4''^7dr^7^)^'^' *^5^ 



dTrx / d'u d'p\ /M 



dTxd. ^/d\v d^M 

^(7u^2 +r;7x^- =0 (7) 



dx \dx- rd(j)dx 

Also by eq. (6) of Art. 3, 



dx dr rd(j)^r ^ ^ 

The cylindrical piece of material is supposed to be of 
such length that the portion to which these equations apply 
is not a.ffected by the manner of application of the couple. 
This portion is, therefore, twisted uniformly from end to 
end; consequently the strain u will not vary with any 
change in x. Hence 

du 

Tx^° (9) 

Eq. (i) then shows that ^ = 0. This was to be antici- 
pated, since a pure shear cannot change the volume or 



856 TORSION IN EQUILIBRIUM. [Ch. 11. 

density. Because ^ = 0, eqs. (2) and (3) at once give 

dp dw p 

-/ = — r7 + -=o (10) 

dr rd(j) r ' ^ 

As the torsion is uniform throughout the portion con- 
sidered, 

— = =— (11) 

Eq. (11), in connection with eq. (10), gives 

d'^w , . 

, ^, -0 (12) 

rdxdcj) 

Eqs. (11) and (12), in connection with eq. (10), reduce 
eq. (5) to the following form: 



% K'5?) 



d^u ,d^u du d .. \ "• / 

^^^dj''^d?'^7dr^''"^d^''^^ dr ' ' ' ^'^^ 

Both terms of the second member of eq. (6) reduce to 
zero by eqs. (9) and (11), and give no new condition. The 
second term of the second member of eq. (7) is zero by 
eq. (9) ; the remaining term therefore gives 

d^w 



As the stress is all shearing, p will not vary with (f). 
Hence 

dp 



Alt. 6.] TORSION IN EQUILIBRIUM. 857 

Eqs. (10), (11), and (15) show that ^=0, and reduce 
eq. (4) to 

div w 

' Eq. (10) now becomes ^ =0, and shows that w does 

not contain 9^; while eq. (14) shows that w does not con- 
tain x' or any higher power of x. The strain w, in connec- 
tion with these conditions, is to be so determined as to sat- 
isfy eq. (16). 

If a is a constant, the following form fulfils all condi- 
tions : 

w = arx (17) 

Eq. (17) shows that the strain w, in the direction of cf), 
i.e., the angular strain at any point, varies directly as the dis- 
tance from the axis of X, and as the distance from the origin 
measured along that axis. This is a direct consequence of 
making Tr<f>=o. 

The quantit}^ a is evidently the angle of torsion, or the 
angle through which one end of a unit of fibre, situated at 
unit's distance from the axis, is twisted ; for if 

r=x = i, w = a. 

An equation of condition relative to the exterior surface 
of the twisted piece yet remains to be determined ; and that 
is to be based on the supposition that no external force what- 
ever acts on the outer surface of the piece. In eqs. (13), 
(14), and (15) of Art. 2, consequently, P -=o. The conditions 
of the problem also make all the stresses except 

T^ = T^r and T^ = T^^ 



858 TORSION IN EQUILIBRIUM. [Ch. II. 

equal to zero, while the cylindrical character of the piece 
makes 

p = go°; .'. cos p=o. 

If cos / be written for cos r, 

cos t -= sin q. 

Eq. (13), just cited, then gives 

Txr cos q + T^^, sin q=o (18) 

But since ^ = o and w = arx, 

and 

Eq. (18) now becomes 

du 

dr dr 



^-=^^57 ^^9) 



Tx<i>=Gl::^,+ar) (20) 



du --^^^^=-r7^' • • • ^''^ 

rd(j) 

in which r^ is the value of r for the perimeter of any normal 
section. 

Eqs. (13) and (21) are all that are necessary and all that 
exist for the determination of the strain u. Eq. (13) must 
be fulfilled at all points in the interior of the twisted piece, 
while eq. (21) must at the same time hold true at all points 
of the exterior surface. 



Art. 6.] TORSION IN EQUILIBRIUM. 859 

After u is determined, Txr and Tx4> at once result from 
eqs. (19) and (20). The resisting moment of torsion then 
becomes 

M =ffT-4> ^' ^'t'-^^'^^ff^-^^^ dcj) + GaIp. (22) 

In this equation Ip= J J r^ .rdcfidr is the polar moment of 

inertia of the normal section of the piece about the axis of 
A^, and the dotible integral is to be extended over the whole 
section. 

According to the old or common theory of torsion 



M=GaIp. 



The third member of eq. (22) shows, however, that such an 
expression is not correct unless tt is equal to zero ; i.e., unless 
all normal sections remain plane while the piece is subjected 
to torsion. It will be seen that this is true for a circular sec- 
tion only. 

It may sometimes be convenient to put eq. (22) in the 
following form: 

M-GJ J rdr.-j^dcp + GaIp = Gj u.rdr + Galp. (23) 
In this equation u is to be considered as 



/ 



'f' du 




di'^'f'' 



while the remaining integration in r is to be so made that 
the whole section shall be covered. 



86o TORSION IN EQUILIBRIUM. [Ch. II. 

The preceding analysis shows that the old or common 
theory of torsion is correct in its expression for torsive 
strain, as it is identical with eq. (17) of Art. 6, i.e., 

w = arx ; 

but it will be seen later that the remaining formulae of the 
common theory are incorrect for all shapes of cross-section 
except the circle. Fortunately the torsion members prin- 
cipally used in engineering practice are shafts of circular 
section. 



Equations of Condition in Rectangular Co-ordinates. 

In the case of a rectangular normal section, the analysis 
is somewhat simplified by taking some of the quantities 
used in terms of rectangular co-ordinates. 

In the notation of Art. 2 all stresses will be zero except 
T3 and T^. Hence eqs. (10), (11), and (12) of that article 
reduce to 

dy dz ' 



^=0; 



dT_ 
dx 

dT\ 

dx 



The strains in the directions of x, y, and z are, respec- 
tively, n, V, and w. Introducing the values of T^ and T.^ 
in the equations above, in terms of these strains, from 
eqs. (11) and (13) of Art. i, and then doing the sam/j.in 
reference to the conditions, 



Art. 6.] TORSION IN EQUILIBRIUM. 86 1 

the following equations will result: 

df^d^^°' ^'^^ 

dv dw . ^ 

dz + d^''° (^7) 

The operations by which these results are reached are 
identical with those used above in connection with semi- 
polar co-ordinates, and need not be repeated. 

Eq. (27) is satisfied by taking 

V = OLXZ ; 
w= — axy ; 

in which a is the angle of torsion, as before. 
Eqs. (11) and (13) of Art. 5 then give 

^.-KI+e)-<^")^ ■ ■ • <=»' 

The element of a normal section is dz dy. Hence the 
moment of torsion is 

M = ffiT,z-T^)dydz; 

.'. M =GJ {zu dz—yu dy) +Galp (31) 

Ip=ff{z'+y')dydz 



862 TORSION IN EQUILIBRIUM. [Ch. II. 

is the polar moment of inertia of any section about the 
axis of A^. 

The integrals are to be extended over the whole section ; 
hence, in eq. (31), zu dz is to be taken as 



z dz. I -rdy 
J -yo dy ^ 



and yn dy as 






dy 

"°du 



dz"^'' 



in which expressions 3/0 and z^ are general co-ordinates of 
the perimeter of the normal section. 

Eq. (26) is identical with eq. (13), and can be derived 
from it, through a change in the independent variables, by 
the aid of the relations 

z=r cos (p and ;y=rsin^. 



Solutions of Eqs. (13) and (21). 

It has been shown that the function u, which represents 
the strain parahel to the axis of the piece, must satisfy 
eq. (13) [or eq. (26)] for all points of any normal section, 
and eq. (21) (or a corresponding one in rectangular co- 
ordinates) at all points of the perimeter ; and those two are 
the only conditions to be satisfied. 

It is shown by the ordinary operations of the calculus 
that an indefinite number of functions u, of r and 0, will 
satisfy eq. (13) ; and, of these, that some are algebraic and 
some transcendental. 

It is further shown that the various functions u which 
satisfy both eqs. (13) and (21) differ only by constants. 



Art. 6.] TORSION IN EQUILIBRIUM. 863 

If u is first supposed to be algebraic in character, and if 
Cp ^2, ^3, etc., represent constant coefficients, the following 
general function will satisfy eq. (13): 

.( c.r sin 6 + c^r^ sin 2(h + cs^ sin 26+ . . .) , , 
( +c'jrcos </) + (;'/^cos 2^ + c'3r^cos 3^+ . . . ) 

and the following equation, which is supposed to belong to 
the perimeter of a normal section only, will be found to 
satisfy eq. (21) : 



— + ^7/ cos <!>■}- c/^ cos 2^ + ^3r^ cos 3^+ . . . 
— c^rsin — c'/^sin 2(j) — c\r- sin t^cJ)— . . . =C. {^^) 

C is a constant which changes only with the form of 
section. 

If -J- and —1-7 be found from eq. (32), w^hile j, be 

taken from eq. (33), and if these quantities be then intro- 
duced in eq. (21), it will be found that that equation is 
satisfied. 

The only form of transcendental function needed, 
among those to w^hich the integration of eq. (13) or eq. (26) 
leads, will be given in connection with the consideration of 
pieces with rectangular section, where it will be used. 

Elliptical Section about its Centre. 

Let a cylindrical piece of material with elliptical normal 
section be taken, and let a be the semi-major and b the 
semi-minor axis, while the angle is measured from a 
with the centre of the ellipse as the origin of co-ordinates, 
since the c^dinder will be twisted about its own axis. The 



864 TORSION IN EQUILIBRIUM. [Ch. 11. 

polar equation of the elliptical perimeter may take the 
following shape: 

7+7*;?T^^°'"^=^MT^- • • • (34) 

By a comparison of eqs. (33) and (34), it is seen that 
c^= / 9 , 7 9x and C = 



and that all the other constants are zero. Hence eq. (32) 
gives 

^ = <^ 2{a' + b'f ^^^ ^"^'^^^ ^^^ ^^' • • (35) 

The quantity represented by / is evident. 
By eqs. (19) and (20) 

r;,;.=6'a: ^2_^^2 ^sin 2^; (36) 



Tt.6=' 



xq> 



""^^(omt^^^^^^^'^v* * • • (37) 



Since "' ^ =dA, A being the area of the ellipse, or 
2 

7ra6, the second member of eq. (22), by the aid of eq. (37), 

may take the form 

M^Gaj d<i>J^ (^^^,r' cos 2 ct> + r'jdr; 
r/b'^ — a^r* r*\ 



Art. 6.] TORSION IN EQUILIBRIUM. 865 

Then using eq. (34), 

If Ip is the polar moment of inertia of the elHpse (i.e., 
about an axis normal to its plane and passing through its 
centre), so that 

nabja^ + b^) 
^P- 4 ' 

then 

A* 

M=Ga^,-j- (39) 

4^'Ip ^-^^^ 

Using / in the manner shown in eq. (35), the resultant 
shear at any point becomes, by eq. (24), 



T=Gar\/p + 2f cos 2^ + 1. 
dT 

gives 

sin 29^ = 0, or (56 = 90° or o®. 

Since / is negative, T will evidently take its maximum 
when (j) has such a value that 2/ cos 2^ is positive, or <j) 
must be 90°. 

Hence the greatest intensity of shear will be found some- 
where along the minor axis. But the preceding expression 
shows that T varies directly as the distance from the centre. 
Hence the greatest intensity of shear is found at the extremities 
of the minor axis. 



866 TORSION IN EQUILIBRIUM. 

Making (/> = 90° and r ^b in the value of T, 



[Ch. 11. 



Taking Ga from eq. (40) and inserting it in eq. (38), 



(40) 



in which 



m 2 "* 



rrab' 

4 



(41) 



or the moment of inertia of the section about the major axis. 

Equilateral Triangle about its Centre of Gravity. 

This case is that of a cyhndrical piece whose normal cross- 
section is an equilateral triangle, and the torsion will be sup- 
posed about an axis passing through b 
the centres of gravity of the different 
normal sections. The cross-section is 
represented in Fig. 3, G being the h 
centre of gravity as well as the origin 
of co-ordinates. 

Let GH = l,GD = a. Then from the 
known properties of such a triangle, 

FD=DB-=BF^2aV~i,. 

Hence the equation for DB is ; r sin — 
Hence the equation for BF is ; 




Fig. 3. 



2a — r cos 



V3 
r cos f a = o 



XT 1 . r r-^ • • 2a — r cos 6 
Hence the equation for FD is; r sm <^ + ,- =0 



Art. 6.] TORSION IN EQUILIBRIUM. 867 

Taking the product of these three equations and reduc- 
ing, there will result for the equation to the perimeter 

— — ^COS^(/)-- — (42) 

2 6a ^ ' 3 ^^ ^ 

Comparing this equation with eq. (33), 

I . ^ 20^ 



Hence 



c, = — ^ and C= — •• 
3 6a 3 



r^ sin S(^ 

«=-«-^^ (43) 



A.nd by eqs. (19) and (20) 



r^sin^^ 
T^r=-CfOL — — ; (44) 



, r^cos 3(/) 
2 a 



') (45) 

Eq. (22) then gives 
M=GaI,-Gaf p^^^drd^; 

^GaU-Gaf'^dr; 

= Ga\Ip—aW~^ =0.6 GaIp = i.S Gaa'Vs; . (46) 
since 7^= polar moment of inertia = 3a ^^3. 



868 TORSION IN EQUILIBRIUM. Ch. II. 

By eq. (24) 

^ ^ I , r^ cos 30 r* 
dT . 



or 

= 0°, 60°, 120°, 180°, 240°, 300°, or 360°. 

The values 0°, 120°, 240°, and 360° make 

cos30=+i; 

hence, for a given value of r, these make T a minimum. The 
values 60°, 180°, and 300° make, 

cos 3(/) = -i; 

hence, for a given value of r, these make T a maximum. 
Putting cos 30 = — I in eq. (47), 



{-?) 



T=Ga[r + ~j. ..... (48) 

This value will be the greatest possible when r is the 
greatest. But = 60°, 180°, and 300° correspond to the nor- 
mal a dropped on each of the three sides of the triangle 
from G. Hence r = a, in eq. (48), gives the greatest intensity 
of shear T.^, or 



m> 



T^--G<^a (49) 



Art. 6.] TORSION IN EQUILIBRIUM. 869 

Or the greatest intensity of shear exists at the middle point 
of each side. Those points are the nearest of all, in the 
perimeter, to the axis of torsion. 

The value of Ga^ from eq. (49), inserted in eq. (46), 
gives 

M=o.4-r^= — -. .... (50) 

a "* 20 ^ 

in which 1= side of section = 2a\/3. 



Rectangular Section about an Axis passing through its 
Centre of Gravity. 

In this case it will be necessary to consider one of the 
transcendental forms to which the integration of eq. (13) 
[or (26)] leads; for if the polar equation to the perimeter be 
formed, as was done in the preceding case, it will be found 
to contain r*, to which no term in eq. (33) corresponds. 

If e is the base of the Napierian system of logarithms 
(numerically ^ = 2.71828, nearly) and A any constant what- 
ever, it is known that the general integral of the partial 
differential eq. (13) may be expressed as follows: 

-ii=^4^«''cos^ ^nVsin^^ . . . . . (51) 

when w^ + w'^ = o; for 

d^u d^u du ^ , , ,,, J. , .L 

dr^ r^ d(j>^ rdr - ^ 

But the second member of this equation is evidently 
equal to zero if 



(w^4-n'^)=o or n'=V — 



870 TORSION IN EQUILIBRIUM. [Ch. 11. 

These relations make it necessary that neither n or n' shall 
be imaginary. 

It will hereafter be convenient to use the following no- 
tation for hyperbolic sines, cosines, and tangents : 



sih i = ; coh t = ; and tah t = -— : 

2 2 e^-\-e~^ 



By the use of Euler's exponential formula, as is well 
known, and remembering that n''^=—n^^ eq. (51) may be 
put in the following form: 

n = i'e^'^cos.^ |-^4^ sin (nr sin <j)) +A\^ cos (nr sin ^)], 

in which the sign of summation is to be extended to all pos- 
sible values of A„ and A\. At the centre of any section for 
which r is zero, u must be zero also, for the axis of the piece 
is not shortened. This condition requires that A\ = o; u 
then becomes 

u = i'^**'' ^°' '^ A^ sin (nr sin ^). 

The subsequent analysis will be simplified by introduc- 
ing the form of the hyperbolic sine, and this may be done 
by adding and subtracting the same quantity to that al- 
ready under the sign of summation, in such a manner that 

u = J[A „ sin (nr sin </>) . sih (nr cos <j)) 

+ l-A „ sin (nr sin 0) e-*"' ^°^ ^]. (5 2) 

Now if the product 

sin (nrsin (p) ^-«^cos^ 



Art. 6.] 



TORSION IN EQUILIBRIUM. 



871 



be developed in a series and multiplied by A,,, one term will 
consist of the quantity 

— r^ sin (j) cos 

multiplied by a constant, and if 

lA^^ sin (nr sin ^) ^-«''cos0 

be replaced by simply, 

— ar^ sin ^ cos ^, 

all the conditions of the problem will be found to be satis- 
fied. This is equivalent to putting 

— ar^ sin ^ cos <j) 

for a general function of r sin <^ and r cos 0. This change will 
give the following form to w, first used by Saint-Venant : 

u = I A „ sin (nr sin cj)) . sih (nr cos </>) — ar^ sin cj) cos 0. (53) 

Fig. 4 represents the cross-section with C as the origin of 
co-ordinates and axis. The angle (p is measured positively 



>^ 



^iC 



Fig. 4. 

from CN toward CH. At the points A^ H, K, and L, in the 



872 TORSION IN EQUILIBRIUM. [Ch. II. 

equation to the perimeter, dr^ will be zero. Hence at those 
points, by eq. (21), 

du 

-J- = ^[A „ sin (nr sin ^) . n cos . coh (nr cos 0) 

4- A „ . w sin <^ . cos {nr sin (j>) . sih (nr cos 9^)] 
— 2ar sin ^ cos =0. 

At the points under consideration <j) has the values 0°, 

90°, 180°, 270°, and 360°. At the points A^ and /v, -o"" or 

180°; hence sin ^ = 0, and both terms of the second mem- 

dii 
ber of -J- reduce to zero, whatever may be the value of n. 

But at H and L, (j) = 90° and 270° ; hence sin = + i or — i 



and cos <^ =0. 



du 



In order, then, that -3- = o at H and L, these must obtain : 



cos nr = cos ( — nr) = o. 
If HL = c and KN = b, then 



nc I nc\ 

cos — =cos ^-— )=o. . . . . (54) 



If the signification of n be now somewhat changed so as 
to represent all possible whole numbers between o and 00 , 
^q- (54) will be satisfied by writing 



2n— I 

TT 



Art. 6.] TORSION IN EQUILIBRIUM. > 873 

for n in that equation. Eq. (53) will then become 

<» ^ . (2n—i . ,\ ., /2W— I \ 

u = 2A^sm. I nr sm ^ I . sih I nr cos I 

— ar- sin cos </) (55) 

The quantity A „ yet remains to be determined by the 
aid of eq. (21), which expresses the condition existing at 
the perimeter of any section. 

Now, for the portion BN of the perimeter, 

h 
r cos -=— > 
2 

dr 
and — ?i will be the tangent of (—</>) , or 
r^d<i> 

dr^ . , .. , . • 









— —-77 = — tan ( 


.-9 


)=U 


Hence 


eq. 


(2 


i) becomes 












du ' 
dr 


= tan 


4>, 




du 
rd^j) 


or 






du 
ar sm 4^ = -r- cos 


4>- 


du 
rdcf> 



(56) 



sin (j). 

Substituting from eq. (55), then making 

r cos =-, 
2 

^^ 2fz-i . (2n—i \ . /2n—i 



sin 0j 



874 TORSION IN EQUILIBRIUM. [Ch. II. 

If r sin <t) be represented by the rectangular co-ordinate 
y, and another quantity by H, the above equation may be 
written 

y^H^ sin'^ + H^ sin ^-^ + H, sin ^ 
c c c 

^_ . /2n—i \ 
+ . . . -H^^smi — - — 7rW+ . . . 



If both sides of this equation be multiplied by 

. /2n-i \ 
sm I Tzyj .ay, 



and if the integral then be taken between the limits o 
and - , it is known from the integral calculus that all terms 

2 

except the n^^' will disappear, and that 



c 



Completing these simple integrations, 



^-((ifVJ^-^^""'-'- 



Hence 

(— i)"-V' 4 2ac 



A 



" (^n-.rn^- c- {2,1-1).- ^^ 



(^-) 



Art. 6.1 



TORSION IN EQUILIBRIUM. 



875 



If this value of A „ be put in eq. (55), and if rectangular 
co-ordinates 

y=rs>m(j) and z=r cos cj) 
be introduced, that equation will become 

u= — azy + 



© 



ac^Z ^ , ; . (57) 



( 2n— 1 \ 
{2n-iyQoh \—^-b) 



This value of u placed in eq. (31) will enable the moment 

of torsion to be at once written. 

c c 
The limits +3^0 and -y^ are +- and , and the limits 

+ ^0 and -Zq are +- and — -. Hence 
2 2 






abc 



.. 271—1. 



{2n — \Y coh 



=^J 



= Q, for brevity; 



-H := 



ahc 



/ 271 I 



2n — I 



ny 



(2W— ij^eoh 



\ 2C 



-.R. 



J 



876 6.] TORSION IN EQUILIBRIUM. 

For the next integration 



[Ch. II. 



f. 



Qz dz = abc 



12 • 



+ 'WFf 



2bc , 2W— I , 4C" ., /2n—i .\ 

_ 7 r-.coh- 7:6—7 ^T~2Sih( — ^ — nb) 



(2W— i)^ coh 



(^•') 



/: 



Ry dy = abc 



■2 /sVc °° 
12 \7tl b 



c" (2yc_^{2n-iyn 



T-^ sih I 7zb 



2C 



(2n— i)^ coh 



2n— I 



2C / ^ 



Thus the integrations indicated in eq. (31) are com- 
pleted. Hence 

M=G'^fQzdz + fRydy + aIpi. . 
Remembering that 



M 






(58) 



But it is known that 



2 n^ 



I (2W— l)* I . 2 .3 ' 2 



5" 



Art. 6.] TORSION IN EQUILIBRIUM. 877 

Hence eq. (58) becomes 



r 64. , tah(i^.6) -[ 



M=Gabc'\ --^j:I ,!_,,, ' |. . (59) 
Since 



I — tah TT I— tah ^TT i— tahs^r 



tah 7z tah ^tt tah t;;r 



■) 



and since 



64 



^-g- = 0.209137, 



and remembering that 

T\2W-i/ ~^~^3^ 5'"^* • * "V 2V295.1215' 
^q- (59) becomes 

M =GabA - - 0.2 10083^ 

/i-tah— i-tah^^— \ 

+ 0.209137^1 ^ + 5 + . . . / 1. (60) 



Eq. (60) gives the value of the moment of torsion of a 
rectangular bar of material. 



878 TORSION IN EQUILIBRIUM. [Ch. 11. 

If z had been taken parallel to b, and y parallel to c, a 
moment of equal value would have been found, which can 
be at once written from eq. (60) by writing b for c and c for b. 

That moment will be 



M--Gacb'\ --0.210083^ 



J. (61 



, / 1 — tah^ I — tah^ 
b\ 20 20 
+ 0.209137 -\^ J + ^1 + 



Eq. (60) should be used when b is greater than c, and eq. 
(61) ivhen c is greater than b, because the series in the paren- 
theses are then very rapidly converging, and not diverging. 
It will never be necessary to take more than three or four 
terms and one, only, will ordinarily be sufficient. The follow- 
ing are the values of 



I — tah — 



for a few values of n: 



1 ^^^\ 
I —tah — 1 =0.083 : 0.00373 : 0.000162 : 0.000007 ; 

n= I : 2 : 3 : 4 

Square Section. 
If c =b either eq. (60) or eq. (61) gives 

M^Gab^ I -- 0.2 101+ 0.209(1 -tah- j ; 

44 
/. M =0.1^06 Gab^-=Ga — ^, . . (62) 



Art. 6.] TORSION IN EQUILIBRIUM. 879 

in which .4 is the area ( = 6^) and Ip is the polar moment of 
mertia i ^^^ )• 

Rectangle in which b = 2C. 
If b = 2C, eq. (60) gives 



M =^Ga. 2C^(- — 0.105 +0.1046 (i — tah k) \; 



I 

3 

A' 
:. M^o.AS! Go^c'---Ga-~-^, . . . (63) 



in which A is the area ( = 2c'^) and /^ = polar moment of in- 
ertia 

J)c' + h'c ^Sc' 
12 6 ■ 

Rectangle in which b = 4.C. 
If 6 ^ 4c, eq. (60) then gives 



M =^Gabc^l — 0.0525 j =1.123 G^ac*; 

A' 
'. M=Ga ^, (64) 

40.2 Ip ■ ^ ^^ 



in which A =area = 4C^ and Ip = polar moment of inertia 

_hc^-\-b^c _i']c^ 
12 .^ 



88o TORSION IN EQUILIBRIUM. fCh. 11. 

If b is greater than 2C, it will be sufficiently near for all 
ordinary purposes to write 

M -^Ga—[i -o.6sj^) (65) 



Greatest Intensity of Shear. 

There yet remains to be determined the greatest inten- 
sity of shear at any point in a section, and in searching for 
this quantity it will be convenient to use eqs. (28) and (29). 

It will also be well to observe that by changing z to y, 
yto — z, c to b, and 6 to c, in eq. (57), there may be at once 
written 



. / 2n—i \ .. /2n— I \ 
u^azy-[-) .ab-I -~ 



(2n— i)^coh 



26 



- ncj 



.(66) 



This amounts to ttirning the co-ordinate axes 90^ 
Since the resultant shear at any point is 



it will be necessary to seek the miaximum of 
du V idu V T' 



The two following equations will then give the points 
desired : 



Art. 6.] TORSION IN EQUILIBRIUM. 



i?) 



du \d^u (du \/ d^u \ ,. , 

+ a.).-. + (rf,-«3'j(5^-«j=o; (67) 



dy \dy jdy 



dz 



du \ dhi \ (du \d^u ,^„. 



It is unnecessary to reproduce the complete substitu- 
tions in these two equations, but such operations show that 
the points of maximum values of T are at the middle points of 
the sides of the rectangular sections, omitting the evident fact 
that r = o at the centre. It will also be found that the great- 
est intensity of shear will exist at the middle points of the 
greater sides. 

This result may be reached independent of any analytical 
test, by bearing in mind that an elongated ellipse closely 
approximates a rectangular section, and it has already been 
shown that the greatest intensity in an elliptical section is 
foimd at the extremities of the smaller axis. 

By the aid of eqs. (28), (29), (57), and (66), it will also 
be foimd that T^^o at the extremities of the diameter c, 
and T^=o at the extremities of the diameter h. The maxi- 
mum value of T will then be 

r„ = -r,= -ff(|-ay)^^^. . . . (69) 



y= 



By the use of eq. (57) 



du 

n . /2n—i \ ^ (2n—i \ 
{^-j)n-x_^^^ ( ny\ .coh( tcz\ 



{2n-iyQ.oh ^-^^^^j 



TORSION IN EQUILIBRIUM. [Ch. II. 

Putting z=-o and y =~ m. this equation, there will result 

T„^Gacr^-^ i_^^__^-|.. (70) 

I '' ' (2W— i)"-^ coll ^ -r.b] I 



If h is greater than c the series appearing in this equation 
is very rapidly convergent, and it will never be necessary to 
use more than two or three terms if the section is square, and 
if h is four or five times c there may be written 

T^=Gac . (71) 



Square Section. 

Making h=c in eq. (70), and making ;z = i, 2, and 3 (i.e., 
taking three terms of the series), there will result 



T 

T^^ =0.6"] 6 Gac; .'. 6^0: = 1.48— ^ 





Inserting this value in eq. (62), 



M = o.2ib'T^=^^^^^ (72) 



M M 

•*• '^rn-O.S-ra^-S^S. .... (73) 



in which 

J b' be 

I = — and a =- =- 

12 22 



Art. 6.] TORSION IN EQUILIBRIUM. 883 



Rectangular Section; h = 2C. 

Making h^2C in eq. (70), and making w = i, only, there 
will result 

T 
T,„-= 0.93(70:^; :. Ga^i.o^-^, 



Inserting this value in eq. (63), 



in which 



i¥ = o.49^^r,„ = i.47-^ (74) 

M M 
.'. r^-=o.68ya-2-3, .... (75) 



^ be' e' ^ c 

I -= — -=7- and a == -. 
126 2 



Rectangular Section; h=/[C. 
Making h = /[c in eq. (70), and making n=-i, only, 

7 

^,« =0.997 Gac; :. Ga = i.oos-^. 

Inserting this value in eq. (64), 

IT 
M = 1.126 r^r,„ -= 1.69 — -. . . . (76) 
m ^ a ' 



. ., M M 

in which 



•. T„,-o.6ya=^o.9^, (77) 



^ be" c^ , c 

y= — =— and a=- 

T2 3 2 



884 TORSION IN EQUILIBRIUM. [Ch. II. 



Circular Section about its Centre. 

The torsion of a circular cylinder furnishes the simplest 
example of all. 

If ro is the radius of the circular section, the polar equa- 
tion of that section is 



— = C (constant). 



Comparing this equation with eq. (33), it is seen that 



^1=^2 = ^3== ••• =^1'=^/= ••• =o- 



By eq. (32) this gives u=o. Hence all sections remain 
plane during torsion. 

Eqs. (19) and (20) then give 

Txr=o and T^4,=Gar (78) 

Eq. (23) gives for the moment of torsion 

M=GaI, (79) 

or 

A ^G 
M =o.K TirQ^ .Ga=- ' 27- Q^> .... (80) 

in which equation A is the area of the section and 

r _^ 



Art. 6.] TORSION IN EQUILIBRIUM. 88$ 

The greatest intensity of shear in the section will be ob- 
tained by making r = ro in eq. (78), or 

T^=Gar,; .\Ga=^^ (81) 



r. 



Eq. (80) then becomes 



i\/ = o.5;rVr^ = 2^ (82)' 



.. ^m =0-64— 3=0. 5-j-ro, (83) 

in which / = — ^• 

4 

It is thus seen that the circular section is the only one 
treated which remains plane during torsion. 



General Observations. 

The preceding examples will sufficiently exemplify the 
method to be followed in any case. Some general conclu- 
sions, however, may be drawn from a consideration of 
eq. (33)- 

If the perimeter is symmetrical about the line from 
which ^ is measured, then r must be the same for + c6 and 
— ^; hence 

c/ =c^^ =c/ = ... =0. 

If the perimeter is symmetrical about a line at right 
angles to the zero position of r, then r must be the same for 

0=9o°+0' and go^-gS'; 



886 TORSIONAL OSCILLATIONS. [Ch. II. 

hence 

^1=^3=^5... =^2'=^/=^/= . . . =0. 

In connection with the first of these sets of results, 
eq. (32) shows that every axis of symmetry of sections repre- 
sented by eq. {^^) will not be moved from its original position 
by torsion. 

If the section has two axes of symmetry passing through 
the origin of co-ordinates, then will all the above constants 
be zero, and its equation will become 



J 4-^2^' COS 2c^ + ^/'cos 4<J^ + ^6^'cos6^+ . . . =K, 



Art. 7. — Torsional Oscillations of Circular Cylinders. 

Two cases of torsional oscillations will be considered, 
in the first of which the cylindrical body twisted is sup- 
posed to be the only one in motion. In the second case, 
however, the mass of the twisted body will be neglected, 
and the motion of a heavy body, attached to its free end, 
will be considered. In both cases the section of the cylin- 
der will be considered circular. 

Since these cases are those of motion, the internal 
stresses are not, in general, in equilibrium; hence equations 
of motion must be used, and those of Art. 3 are most con- 
venient. Of these last, the investigations of the preceding 
article show that eq. (4) is the only one which gives any 
conditions of motion in the problem under consideration. 

Putting the value of 



Art. 7.] TORSIONAL OSCILLATIONS, 887 

in eq. (4) of Art. 3, that equation may take the form 
d'^w G dhu d\v .^d^w 

For brevity, 6^ is written for ~. 

That dimension of the cross-section of the body which 
lies in the direction of the radius will be assumed so small 
that w may be considered a function of x and t only. The 
results will then apply to small solid cylinders and all hollow 
ones with thin walls. 

The general integral of eq. (i), on the assumption just 
made, is (Booles' ''Differential Equations," Chap. XV, 
Ex. i) 

w=^j{x\y:)^F{x-ht), 

in which / and F signify any arbitrary functions whatever. 
Now it is evident that all oscillations are of a periodic char- 
acter, i.e., at the end of certain equal intervals of time, w 
will have the same value. Hence since / and F are arbitrary 
forms, and since circular functions are periodic, there may 
be written 

w -^AJ sin (a .^^x + a ^.bt) + sin (a ,^x — a ,. bt) ] 

-BJcos {a ^^x + a ^.bt)- cos (a^x-a^pt)], (2) 

in which a^, A^, and S,^ are coefficients to be determined. • 
Substituting for the sines and cosines of sums and differ- 
ences of angles, 

w = 2 sin a ^^x( A ^. cos a ^bt + B^^ sin a Jjt). . . (3) 

Let the origin of co-ordinates be taken at the fixed end 
of the piece, w must then be equal to zero, as is shown by 



888 CIRCULAR CYLINDERS. [Ch. II. 

eq. (3). But there may be other points at which w is always 
equal to zero, whatever value the time t may have. These 
points, called nodes, are found by putting w; = o, or 

sin aji; = o (4) 

This equation is satisfied by taking 

^«^a' a"' T' • * • ' "a * 

and x=a] in which a is the length of the piece. 
Hence at the distances 

a a a 

from the fixed end of the piece, there will exist sections which 
are never distorted or moved from their positions of rest. These 
are called nodes, and one is assumed at the free end, although 
such an assumption is not necessary, since a is really the 
distance from the fixed end to the farthest node and not 
necessarily to the free end. 

If, as is permissible, An and B^ be written for twice 
those quantities, the general value of w now becomes 

. nxf ^ Tcht ^ . 7tht\ 

w = sm — M., cos — + S, sm — I 

a\ ^ a ^ a J 

. 27ZX/ ^ 27:bt _ . 2Kbt\ 

+ sm ^—[ A^ cos ^^ — +B. sm ~ — I 
^ a \ ^ a ^ a J 



. nizxl , nizbt ^ . n7:bt\ , ^ 

+ sm (An cos +Bn sm ). . . (5) 

. a \ a a / 



Art. 7.] TORSIONAL OSCILLATIONS, 889 

The coefficients A and B are to be determined by the 
ordinary procedure for such cases. Let 

be the expression for the initial or known strain at any point, 
for which the time / is zero. Then if An is any one of the 
coefficients Ay 

A 2 /*^, . , . nnx . ' ,,^ 

"^ayo^^^^^"^ .... (6) 

The velocity at any point, or at any time, will be given 



by 



dw . nxf ^ . Tzht ^ 7tht\7tb 

-sm — lA, sm — — 5, cos I — -. . . (7) 



In the initial condition, when the time is zero, or ^=0, 
it has the given, or known, value 

dw, ^, . 7th( ^^ . nx ^ . 2nx ^ . -inx \ 

Then, as before, 

^«==-A / <^(^)sin dx (8) 

Thus the most general value of w is completely deter- 
mined. 

The intensity of shear at any place or time is given by 



dw 
dx 



w being taken from eq. (5). 



Sgo CIRCULAR CYLINDERS. [Ch. II. 

The second case to be treated is that of the torsion pen- 
dukini, in which the mass of the twisted body is so incon- 
siderable in comparison with that of the heavy body, or 
bob, attached to its free end that it may be neglected. 

Let AI represent the mass of the pendulum bob, and k 
its radius of gyration in reference to the axis about which it is 
to vibrate, then will Mk- be its moment of inertia aboiit the 
same axis. 

The unbalanced moment of torsion, with the angle of 
torsion a, is, by eq. (9) of Art. 6, 

Galp. 

The elementary quantity of work performed by this 
unbalanced coujjle, if /? is the general expressiorl for the 
angular velocity of the vibrating body, is 

Galp.^dt. 

This quantity of energy is equal in amount but opposite 
in sign to the indefinitely small variation of actual energy 
in the bob ; hence 



Gal^Sdt =-d (^^^^) = -Mk'pd^, 



But if a is the length of the piece twisted, 
d(aa) dHaa) 



^)(.a) = -M.^^. 



Art. 7.] TORSIONAL OSCILLATIONS. 89 1 

Multiptying this equation by 2d{aa), and for brevity 
putting 



{~^)--H, {Mk')=K: 



then integrating and dropping the common factor a^ 

When a=a^, the value of the angle of torsion at the 
extremity of an oscillation, the bob will come to rest and 

-J- will be zero. Plence 
at 



C = Ha,^ 



and 



da \H .^ 

/. sin-^— =^J^ + (C'=-o). ... (9) 

C =0 because a: and ^ can be put equal to zero together. 
At the opposite extremities of a complete oscillation a 
will have the values 

( + «,) and (-a^). 
Putting these values in the expression 



^ = V^.sin-- ( 



10) 



892 TORSION PENDULUM. [Ch. II. 

and taking the difference between the results thus obtained, 
the following interval of time for a complete oscillation will 
be found: 



=^^l7/=^ 






The time required for an oscillation is thus seen to vary 
directly as tJie square root of the moment of inertia of the hob 
and the length of the piece, and inversely as the square root of 
the coefficient of elasticity for shearing and the polar moment 
of inertia of the normal section of the piece twisted. 

The number of complete oscillations per second is -. If 

this number is the observed quantity, the following equa- 
tion will give G : 



=0) 



Ip 



The formulas for this case should only be used when the 
mass of the cylindrical piece twisted is exceedingly small in 
comparison with M. 

Art. 8. — Thick, Hollow Spheres. 

In order to investigate the conditions of equilibrium of 
stress at any point within the material which forms a thick 
hollow^ sphere, it will be most convenient to use the equa- 
tions of Art. 4. As in the case of a thick hollow cyhnder, 
the interior and exterior surfaces of the sphere are supposed 
to be subjected to fluid pressure. 

Let r' and r^ be the interior and exterior radii, respec- 
tively. 

Let — p and — p^ be the interior and exterior intensities, 
respectively. 



Art. 8.] THICK, HOLLOIV SPHERES. 893 

Since each surface is subjected to normal pressure of uni- 
form intensity no tangential internal stress can exist, but 
normal stresses in three rectangular co-ordinate directions 
may and do exist. Consequently, in the notation of Art. 4, 

With a given value of r, also, a uniform state of stress 
will exist. Neither N^ nor N<f> can, then, vary with cj) or ([/. 
By the aid of these considerations, and after omitting R^, 
^0, ^0, and the second members, the eqs. (i), (2), and (3) 
of Art. 4 reduce to 

dNr ^ 2Nr-N^-N^ 

-5r+ r =^', •.•.(!) 

-A^^ + A^^=o (2) 

By eq. (2) 

N^=N^. 

Eq. (i) then becomes 

dNr Nr-N^ 

-5/ + ^— r-=- (3) 

On account of the existing condition of stress which has 
just been indicated it at once results that 

lfj = CO=0, 

and that ^ is a function of r only. 

Eqs. (4) to (10) of Art. 4 then reduce to 

'-i-'i-' (4) 

"'-^r'-"^!--- • • • . . (5) 



Nt=Nt = ^^d + 2G^. 



I-2C ■ r (6) 



894 THICK, HOLLOW SPHERES. [Ch. II. 

After substitution of these quantities, eq. (3) becomes 

2Gt ((Pp 2rdp—2pdr\ d^p dp p 

T^^ [d^ + ?~dF~) + '^dr^ + ^^'7d~^^? '-^""^ 

or 



d'p K'^'r 

— o. 



dr^ dr 
One integration gives 



dp 2p ^ 



Hence 6, the rate of variation of volume, is a constant 
quantity. Eq. (7) may take the form 

r dp-{- 2p dr = cr dr. 

As it stands, this equation is not integrable, biit, by in- 
specting its form, it is seen that r is an integrating factor. 
Multiplying both sides of the equation, then, by r, 

r^dp+ 2rpdr =d(r^p) =cr'^dr; 

T ^ CT b 

:. r'p = c-+b; .-. P=- + ~2' ... (8) 

Substituting from eqs. (7) and (8) in eq. (5), 

__ 2GV 2Gc 4bG , ^ 

A^r=7z:^^+— - 7f; .... (9) 

It is obvious what A represents. 



Art. 8.] THICK, HOLLOIV SPHERES. . 895 

When / and r^ are put for r, A^^ becomes —p and —p^. 
Hence 



and 






These equations express the conditions involved in eqs. 
(13), (14), and (15) of Art. 2. 
The last equations give 



ip,-p)ry\ 
r'^ — r^ 

• • -^ ^/3 ^ 3 • 



4^^=~^3_^3 » 



These quantities make it possible to express A^^ and .Y^ 
independently of the constants of integration, c and 6, for 
those intensities become 

_ P/ ,'-pr'' _ (p,-p)ry' I . 

^^r — ^3_^ 3 /3_^ 3 '^31 • V-tO/ 

Thus it is seen that N^=N4, has its greatest value for 
the interior surface ; that intensity will be called h. 

It is now required to find r^ — r'=tin terms of h, py and /?j. 
If r =r' in eq. (11). 



896 THICK, HOLLOfV SPHERES. [Ch. II. 

Dividing this equation by r'^ and solving, 





/3 


2(h + p) 
2h-p + 3Pi' 


^- 


,3 2{h+p) 



-/. . . (12) 



If the intensities p and p^ are given for any case, eq. (12) 
will give such a thickness that the greatest tension h (sup- 
posing p^ considerably less than p) shall not exceed any 
assigned value. If the external pressure is very small com- 
pared with the internal, ^^ may be omitted. 

The values of .4 and 4Gb allow the expressions for c and 
b to be at once written. 

If p^ is greater than p, nothing is changed except that 
A^^ =A^^ becomes negative, or compression. 



CHAPTER III. 

THEORY OF FLEXURE. 
Art. 9. — General Formulae. 

If a prismatic portion of material is either supported at 
both -ends, or fixed at one or both ends, and subjected to 
the action of external forces whose directions are normal 
to, and cut, the axis of the prismatic piece, that piece is said 
to be subjected to " flexure." If these external forces have 
lines of action which are oblique to the axis of the piece, it 
is subjected to combined flexure and direct stress. 

Again, if the piece of material is acted upon by a couple 
having the same axis with itself, it will be subjected to '' tor- 
sion." 

The most general case possible is that which combines 
these three, and some general equations relating to it will 
first be established. 

The co-ordinates axis of X will be taken to coincide with 
the axis of the prism, and it will be assumed that all external 
forces act upon its ends only. Since no external forces act 
upon its lateral surface, there will be taken 

retaining the notation of Art. 2. These conditions are not 
strictly true for the general case, but the errors are, at most, 
excessively small for the cases of direct stress or flexure, or 

897 



898 THEORY OF FLEXURE. [Ch. III. 

for a combination of the two. By the use of eqs. (12), (21), 
and (22) of Art. i the conditions just given become 



r /du dv dw 
i — 2r \dx- dy dz 



'- + ^+=^)+| = o; ... (I) 



r /du dv dw \ dw 
T=^VS + 5^+^/+5F^°' • • • ^^^ 

dv dw , . 

dF + d^^°- (3) 

Eqs. (i) and (2) then give 

dv dw 

d^-d^=° (4) 

In consequence of eq. (4) eqs. (i) and (2) give 

dv _dw du 

dy dz dx ^^^ 

By the aid of eq. (5) and the use of eqs. (11), (13), and 
(20) of Art. I, in eqs. (10), (11), and (12) of Art. 2 (in this 
case Xq = Yq=Zq = o), there will result 

d'^u d^u d^u .^. 

d^u d^v 
d^^d^' ^''' ••••••• (7) 

d^u d^w . 

d^z^di^^"" ^^^ 

Eqs. (3), (5), (6), (7), and (8) are five equations of 
condition by which the strains u, v, and w are to be deter- 
mined. 



Art. 9.] GENERAL FORMUL/E. 899 

Let eq. (6) be differentiated in respect to x: 

dhi d^u d^u 

dx^ dy^ dx dz^ dx 

From this equation let there be subtracted the sum of 
the results obtained by differentiating eq. (7) in respect to y 
and (8) in respect to z: 

d^u d^v dhv 

dx^ dx^ dy dx^ dz 

In this equation substitute the results obtained by 
differentiating eq. (5) twice in respect to x, there will result 

.3 HP 

d^u \dx^ 

"3J'=^S^ =°- W 

This result, in the equation immediately preceding eq. 
(9), by the aid of eq. (5) will give 

d'v 

o. 



dx^ dy 

After differentiating eq. (7) in respect to y, and substi- 
tuting the value immediately above, 

d^u _ \dxi 
dy'Tx 57" =° ('°) 

Eqs. (9) and (10) enable the second equation preceding 
eq. (9) to give 

^du" 



d^u _ \dx^ 
dz^x dz' ^° ^^'^ 



goo THEORY OF FLEXURE, [Ch. III. 

Let the results obtained by differentiating eq. (7) in 
respect to z and (8) in respect to y be added : 

d^u ■ d^v d^w 
dx dy dz dx^ dz dx^ dy 

The sum of the second and third terms of the first mem- 
ber of this equation is zero, as is shown by twice differentiat- 
ing eq. (3) in respect to x. Hence 

d^u \dx, , . 

= 0. .... . (12) 



dy dz dx dy dz 
Eqs. (9), (10), (11), and (12) are sufficient for the 

d'VL 

determination of the form of the ftinction -j-, if it be assumed 
to be algebraic, for 



Eq. (9) shows that x"^ does not appear in it ; 
" (10) - - f 



" ^,2 << <■<■ 



(11) 


(( 


<( 


2:^ 


<( 


(< 


(( 


(' 


(12) 


(< 


(( 


yz 


(< 


(( 


(( 


ii 



The products xz and xy may, however, be foimd in the 
fimction. Hence if a, a^, a^, b, h^, and h^ are constants, 
there may be written 

du 

— =a + a^z-\-a^y + xib + b,z + b^y). . . . (13) 

"Eq. (5) then gives 

dv div 

jy = :^ = -r{a + a,z + a^y + x(b + b^z + b^)], . (14) 



Art. 9.] GENERAL FORMUL/E. 9°! 

Substituting from eq. (13) in eqs. (7) and (8), 

^2 ==-a^-h^\ (15) 

^ = -a,-b,x (16) 

The method of treatment of the various partial deriva- 
tives in the search for eqs. (13) and (14) is identical with that 
given by Clebsch in his " Theorie der Elasticitat Fester 
K or per." 

It is to be noticed that the preceding treatment has been 
entirely independent of the form of cross-section or direction 
of external forces. 

It is evident from eqs. (13) and (14) that the constant a 
depends upon that component of the external force which 
acts parallel to the axis of the piece and produces tension or 
compression only. For (pages 9, 10) it is known that 
if a piece of material be subjected to direct stress only, 

du ^ dv dw 

-J- =-a and t~ = j~ = — ^^ ; 
dx dy dz 

the negative sign showing that ra is opposite in kind to a, 
both being constant. 

Again, if z and y are each equal to zero, eq. (13) shows 
that 

du . 

-T~ =a-\-bx. 
dx 

Hence hx is a part of the rate of strain in the direction of x 
which is uniform over the whole of any normal section of the 
piece of material, and it varies directly with x. But such a 



902 THEORY OF FLEXURE. [Ch. III. 

portion of the rate of strain can only be produced by an 
external force acting parallel to the axis of X, and whose 
intensity varies directly as x. But in the present case 
such a force does not exist. Hence h must equal zero. 

The eqs. (13), (14), (15), and (16) show that a^, h^ and 
Oj, 62 are symmetrical, so to speak, in reference to the co- 
ordinates z^indy, while eqs. (13) and (14) show that the nor- 
mal intensity A^^ is dependent on those, and no other, con- 
stants in pure flexure in which a = o. It follows, there- 
fore, that those two pairs of constants belong to the two 
cases of flexure about the two axes of Z and Y. 

No direct stress N^ can exist in torsion, which is simply a 
twisting or turning about the axis of X. 

Since the generality of the deductions will be in no man- 
ner affected, pure flexure about the axis of Y will be con- 
sidered. For this case 

a=a^ = h^ = o =b. 
Making these changes in (13) and (14), 

du 

^=a,z + b,xz; (17) 

dv _ dw du 

^~5F = -'•5^ = -''(«.« + V^)- • • • (18) 

. . du dv dw 

■■ ^^Tx+d^ + d^-'(^^ + b^^^(^-^r)-- ■ (19) 



Also, 



^ i~2r dx 

N, = 2G{r-\- i)(a^ + b^x)z=E(a^+b,x)z, . (20) 



Art. 9.] GENERAL FORMUL/E. 903 

since 

2G{r-^i)=E, 

Taking the first derivative of N^^ 

^'=£(a, + V) (21) 

This important equation gives the law of variation of 
the intensity of stress acting parallel to the axis of a bent 
beam, in the case of pure flexure produced by forces exerted 
at its extremity. That equation proves that in a given nor- 
mal section of the beam, whatever may be the form of the 
sectiCfn, the rate of variation of the normal intensity of stress is 
constant ; the rate being taken along the direction of the external 
forces. 

It follows from this that A^^ must vary directly as the 
distance from some particular line in the normal section 
considered in which its value is zero. Since the external 
forces F are normal to the axis of the beam and direction 
of A/'j, and because it is necessary for equilibrium that the 
sum of all the forces N^dy dz, for a given section, must be 
equal to zero, it follows that on one side of this line tension 
must exist, and on the other compression. 

Let A^ represent the normal intensity of stress at the 
distance unity from the line, h the variable width of the 
section parallel to 3/, and let i = hdz. The sum of all the 
tensile stress in the section will be 



[ NzA=n\ 

Jo J c 



zJ. 

The total compressive stress will be 

Np zJ. 

J -zi 



904 



THEORY OF FLEXURE. 



[Ch. III. 



The integrals are taken between the limits o and the greatest 
value of z in each direction, so as to extend over the entire 
section. In order that equilibrium may exist, therefore, 





Fig. I. 



n\\ zA^r za\ =0. 



■•■1: 



zA =0. 



(22) 



Eq. (22) shows that the line of no stress must pass through 
the centre of gravity of the normal section. 

This line of no stress is called the neutral axis of the 
section. Regarding the whole beam, there will be a sur- 
face which will contain all the neutral axes of the different 
sections, and it is called the neutral surface of the bent 
beam. The neutral axis of any section, therefore, is the 
line of intersection of the plane of section and neutral sur- 
face. 

Hereafter the axis of X will be so taken as to traverse 
the centres of gravity of the different normal sections 
before flexure. The origin of co-ordinates will then be 



Art. 9 



GENERAL FORMUL/E. 



90^ 



taken at the centre of gravity of the fixed end of the beam, 
as shown in Fig. i. 

The value of the expression {a^ + h^x), in terms of the 
external bending moment, is yet to be determined. Con- 
sider any normal section of the beam located 
at the distance x from 0, Fig. i, and let 
OA =1. Also, let Fig. 2 represent the sec- 
tion considered, in which BC is the neutral 
axis and d^ and d^ the distances of the most 
remote fibres from BC. Let moments of all 
the forces acting upon the portion (/ — x) of 
the beam be taken about the neutral axis BC. If, again, h 
is the variable width of the beam, the internal resisting 
moment will be 




Fig. 



rd' rd' 

NJjzdz=E(a^ + b^x)\ z\hdz, 

J —di J —di 



But the integral expression in this equation is the moment 
of inertia of the normal section about the neutral axis, which 
will hereafter be represented by I. The moment of the 
external force, or forces, F, will be F{l — x), and it will be 
equal, but opposite in sign, to the internal resisting moment 
Hence 



F l-x)=M=-E(a^ + b^x)L 



(23) 



(a^ + b^x) = 



K 
EF 



(24) 



Substituting this quantity in eq. (16), 



dhv_M_ 
dx' "EF 



(25) 



9o6 THEORY OF FLEXURE, [Ch. IIL 

It has already been seen (page 38) that eq. (25) is one 
of the most important equations in the whole subject of the 
"Resistance of Materials." 

An equation exactly similar to (25) may of course be 
written from eq. (15); but in such an expression M will 
represent the external bending moment about an axis par- 
allel to the axis of Z. 

No attempt has hitherto been made to determine the 
complete values of ti, v, and w, for the mathematical opera- 
tions involved are very extended. If, however, a beam be 
considered ^vhose width, parallel to the axis of Y, is indefi- 
nitely small, u and w may be determined without difficulty. 
The conclusions reached in this manner will be applicable 
to any long rectangular beam without essential error. 

If y is indefinitely small, all terms involving it as a factor 
will disappear in ti and w ; or, the expressions for the strains u 
and w will he functions of z and x only. But making u and w 
functions of z and x only is equivalent to a restriction of 
lateral strains to the direction of z only, or to the reduction 
of the direct strains one half, since direct strains and lateral 
strains in two directions accompany each other in the un- 
restricted case. Now as the lateral strain in one direction 
is supposed to retain the same amoimt as before, while the 
direct strain is considered only half as great, the value of 
their ratio for the present case will be twice as great as that 
used on pages 9 to 12. Hence 2r must be written for r, in 
order that that letter may represent the ratio for the unre- 
stricted case, and this will be done in the following equations. 

Since w and u are independent of y, 

dw du dv 

1— =-3- =0, and I^=G-r-' 
dy dy ' ^ dx 

But, by eq. (14), 

V = — 2r(ai + h^x)zy + f{x, z). 



Art. 9.] GENERAL FORMULy^, 907 

By eq. (3), since 

dw 

^ = -2ria^ + \x)y + ^J(x, z) =0. 

This equation, however, involves a contradiction, for it 
makes f(x, z) equal to a function which involves y, which is 
impossible. Hence 

f{x,z)^o. 

Consequently 

^^ / 7 s 

which is indejfinitely small compared with 

^=-2r{a^ + b,x)z, 

and is to be considered zero 
Because f(x, z) = o, 

dv . 

This quantity is indefinitely small; hence 

Tg = — 2Grh^zy 

is of the same magnitude. 

Under the assumption made in reference to y, there may 
be written, from eqs. (17) and (18), 

u = a^xz + h^-z-\-f{z); .... (26) 
w=-r{a^z^-\-h^xz'')-[-f{x), . . . (27) 



9o8 THEORY OF FLEXURE. [Ch. III. 

Using eq. (26) in connection with eq. (6), 

By two integrations, 

f'{z)^-^-c'z + c" ' (28) 

Using eq. (27) in connection with eq. (8), 

By two integrations, 

The functions u and w now become 



u = a^xz + b^-—z '—-c'z + (/^; . . . (29) 

^ 3 



w= — ra^z^ — rb^xz^ — b^~ ^ — + c^x + (7^. . (30) 

The constants of integration c' , c" , etc., depend upon 
the values of u and w, and their derivatives, for certain 
reference values of the co-ordinates x and z, and also 
upon the manner of application of the external forces, F, at 
the end of the beam, Fig. i . The last condition is involved 
in the application. of eqs. (13), (14), and (15) of Art. 2. 



Art. 9.1 GENERAL FORMULA. 909 

In Fig. I let the beam be fixed at 0. There will then 
result, f or :^; = o and z=o, 



(du \ 



2 = 0< 

x = o 



(u = o, and w = o) 

In virtue of the last condition, 

c =^11=0- 
In consequence of the first, 



After inserting these values in eqs. (29) and (30), 
du ^ x^ . . 



dz 
dw 



= — rh^z^ — 6j — — a^x + c^. 



^^=^Gt+S = -^^(^+^>^+^'^.- • (31) 



The surface of the end of the beam, on which F is applied, 
is at the distance I from the origin and parallel to the 
plane ZY . Also, the force F has a direction parallel to the 
axis of Z. Using the notation of eqs. (13), (14), and (15) of 
Art. 2, these conditions give 

cos^ = i, cosg = o, cosr=o, 

C0S7r=0, COS/ = 0, COS|0 = I. 



9IO THEORY OF FLEXURE, [Ch. III. 

Since, for x = l, 

M = F(l-x)=o, 

eqs. (24) and (20) give N.^=o for all points of the end sur- 
face. Eq. (15) is, then, the only one of those equations 
which is available for the determination of q. 
That equation becomes simply 

For a given value of z, therefore, any value may be as- 
sumed for 72- For the upper and lower surfaces of the beam 
let the intensity of shear be zero; or iov z= ±d let T^ = o. 
Hence, by eq. (31), 

c,=b,(i+r)d\ 

Fh 
.-. T,^^{d'-z^). ...... (32) 

The constants a^ and b^ still remain to be found. The 
only forces acting upon the portion (/ — x) of the beam are 
F and the sum of all the shears T^ which act in the section x. 
Let Jy be the indefinitely small width of the beam, which, 
since z is finite, is thus really made constant. The princi- 
ples of equilibrium require that 

r T,.Jy.dz=Gb,(i+r)r {d\ Ay .dz-z\ Ay .dz) =F. 

The first part of the integral will be 2 Ayd^, and the second 
part will be the moment of inertia of the cross-section (made 



Art. 9-1 GENERAL FORM UL/E. 911 

rectangular by taking Ay constant) about the neutral axis. 
Hence 

77 77 

2Gh,{i+r)I=F, or ^ - ,g(i +^)/ °gj- • '^H) 

.: T.^Yl^d'-^') (34) 

If «; = o in eq. (24), 

^^.= -£7 (35) 

Thus the two conditions of equilibrium are. involved in 
the determination of a^ and h^. The complete values of the 
strains u and w are, finally, 

F I x^ _z^ 
2 3 



^ = gjUT"T-^^^)' • • (36) 



w 



F / :r^ Z:r^\ F(Px 

= -(^lr,^-rx,^--+-)+-^. . . (37) 



These results are strictly true for rectangular beams of 
indefinitely small width, but they may be applied to any 
rectangular beam fixed at one end and loaded at the other, 
with sufficient accuracy for the ordinary purposes of the 
civil engineer. It is to be remembered that the load at the 
end is supposed to be applied according to the law given 
^y ^q.- (34) » ^ condition which is never realized. Hence 
these formulas are better applicable to long than short 
beams. 



912 THEORY OF FLEXURE, [Ch. III. 

The greatest value of T^, in eq. (34), is found at the 
neutral axis by making z = o\ for which it becomes 

^^="2r=f^- • • • • • (38) 

—J is the mean intensity of shear in the cross-section; 

hence the greatest intensity of shear is once and a half as 
great as the mean. 

In eq. (36), if 2 = 0, u = o. Hence no point of the neu- 
tral surface suffers longitudinal displacement. 

In eq. (37) the last term of the second member is that 
part of the vertical deflection due to the shear at the neu- 
tral surface, as is shown by eq. (38). The first term of 
the second member, being independent of <r, is that part 
of the deflection which arises wholly from the deformation 
of the normal cross-section. 

The usual modification of this treatment, designed to 
supply formulae for the ordinary experience of the engineer, 
has already been given in preceding articles. 



APPENDIX II. 

CLAVARINO'S FORMULA, 

In Art. 13 reference is made to Clavarino's formula 
for thick cylinders. It will be sufficient here to establish 
the equation for the circumferential or hoop tension in 
a thick cylinder to illustrate Clavarino's fundamental idea. 

If / represents the unit strain in the direction of a 
tensile force acting alone and whose intensity is T, and 
if V is the unit longitudinal strain in the same direction 
under the same stress T but with two intensities of com- 
pressive stress R and 5 acting at right angles to each other 
and to the stress T with corresponding direct unit strains h 
and hy and finally if r is the ratio of the lateral strain 
divided by the direct or longitudinal strain, then will 

r=/-fr/i+r/2 (i 

According to Clavarino's view a lateral strain repre- 
sents the action or an actual force or stress with an in- 
tensity equal to the modulus of elasticity E multiplied 
by the lateral unit strain. Consequently he considered 

El' = r = T-\-rR + rS y2) 

In the case of the thick cylinder T is the intensity of 
stress originally established by Lame and given by eq. 
(16) Art. 5 of Appendix I, while R is the radial compres- 
sion given by eq. (15) of the same Art., and 5 is the intensity 

913 



914 C LAV A RING'S FORMULA. App. II. 

of longitudinal tensile stress existing if the cylinder has 
closed ends and it is found by eq. (3) ; 

^ ri2-/2 ^3J 

As 5 is a tensile stress and causes a negative lateral 
strain the term r5 in eq. (2) must have the negative sign. 
Again, eq. (15), Art. 5, of Appendix I is so written as to 
make R negative. Hence, for the present purpose, eq. (2) 
must be written : 

EV=r = T-rR-rS (4) 

Substituting the values of R and T from eqs. (15) and 
(16), Art. 5, Appendix I, and the value of 5 from eq. (3), 
in eq. (4) and taking r = \, 

r=(p,n^-pr'^+4{p,-pY-^)j^^r^y ■ (s) 
If r = r' in eq. (5), the greatest value of T' becomes: 

r = \{r'^+4ri')p-5Piri'] . / ,,y . . (6) 



Finally, if ^1 =0, 






If the stress 5 = o the corresponding modifications of 
the formulae are obvious. 

Eq. (6) gives for the exterior radius; 






App. II.] CLAVARINO'S FORMULA. 915 

These equations illustrate Clavarino's formulae. For 
the reasons given fully in Art. 13, they can be considered 
approximate only. 

Related closely to Clavarino's method is that procedure 

of arbitrarily assuming T -\ — = constant in an analysis of 

3 
the stresses in the wall of a thick cylinder. At best the 
results are but approximate. 



APPENDIX III. 

RESISTING CAPACITY OF NATURAL AND 
ARTIFICIAL ICE. 

In the early part of 19 13 two graduating students in 
Civil Engineering, Messrs. A. F. Lipari and R. M. Marx, 
at Columbia University, acting under the immediate direc- 
tion of Mr. J. S. Macgregor, in charge of the testing labora- 
tory of the Department of Civil Engineering, conducted a 
series of physical tests of natural and artificial ice, both in 
compression and in flexure. These tests were made with 
scrupulous care as to the application of loads to test pieces 
and in the quantitative determination of results. The 
test pieces in compression were subjected to their loads in 
the cooling apparatus employed. The compression tests of 
the natural ice were made with the load applied in some 
cases normal to its natural surface and in other tests parallel 
to that surface, in other words normal to its bed and parallel 
to its bed. 

The behavior of the two kinds of ice in the tests was 
quite different in some respects. A block of clear artificial 
ice would soon be clouded under a gradual application of 
loading by the formation of crystals, which finally would 
determine the lines of compressive failure; while the ten- 
dency of the natural ice was to separate and fail in columns. 
In both cases, however, there was a distinct tendency to 
shear on oblique planes, making an angle of about 45° with 
the direction of loading. The separation along these shear 
planes was distinctly marked in many specimens. 

916 



App. III. 



RESISTING CAPACITY OF ICE. 



917 



In general the height of the compression test specimens 
was about twice the greatest cross dimensions, but the larg- 
est specimens tested were exceptions to this observation. 
The accompanying table gives a concise statement of the 
results of the fifty-seven tests of natural ice in compression 
and of the thirty-one compressive tests of the artificial ice. 

Table I. 

NATURAL ICE IN COMPRESSION. 



Size of Test Pieces. 


Number of 
Tests. 


Ult. Comp. Resistances 
Pounds per Sq. In. 




Max. 


Mean. 


Min. 


3.25 ins. by 3.75 ins. 

to 
9.8 ins. by 13.9 ins. 


57 


II32 


543 


100 



ARTIFICIAL ICE IN COA/[PRESSION 



3 

10.5 


ins. by 3.2 ins. 

to 
ins. by 10.2 ins. 


31 


368 


185 





The dimensions of the cross-sections of the test pieces 
are seen to vary greatly. The number of pieces tested with 
the larger cross-sections was not enough to establish any 
definite relation between the ultimate compressive resist- 
ances per square inch and the areas of the cross-sections of 
the test pieces. Within the limits of these tests there ap- 
pears to be little, if any, material variation of ultimate 
resistance with the increase of cross-section. 

It is important to observe that the ultimate resistance 
of the artificial ice is much less than that of the natural. 
In fact, the mean ultimate resistance of the natural ice is 
nearly three times as great as the mean ultimate resistance 



9l8 RESISTING C/IPACITY OF ICE. [App. III. 

of the artificial, and about the same relation holds for the 
maximum intensities. 

The temperature of the test pieces as determined by 
thermo-couples during the actual procedure of testing ranged 
generally from about +28° Fahr. to about freezing. It is 
probable that the temperature of the ice was considerably 
lower than indicated by the apparatus. 

The test pieces were not selected with any special care, 
but were fair averages of natural and artificial ice as or- 
dinarily sold in quantities for the usual purpose of city 
consumption. Naturally the quality varied materially in 
many blocks as bought, causing correspondingly wide 
variations in the ultimate resistances determined. The 
results of these compressive tests show that sound natural 
ice at about the temperatures indicated may be expected 
to give on the average an ultimate resistance of about 
500 lbs. per sq. in., with a range of perhaps 100 to 1000 
lbs. per sq. in. The artificial ice tested appears to have 
had about one-third the ultimate resistance only of the 
natural ice. 

In some cases the test pieces of natural ice appeared to 
give somewhat greater ultimate resistances when tested 
on their beds than when tested on edge. In scrutinizing 
the whole list, however, there appears to be but little, if any, 
difference. Hence no distinction of this kind has been made 
in Table I, but all the tests have been treated as of one 
group. 

Table II shows the results of testing beams of both 
natural and artificial ice with loads applied at the centre 
of span. The effective span in all cases was 18 inches. 
The normal cross-sections of the beams were square and 
varied but little from 3.5 inches by 3.5 inches. There 
were nine such tests of beams of natural ice and twelve of 
beams of artificial. ice. The modulus of rupture is the usual 



App. III.] 



RESISTING CAPACITY OF ICE. 



919 



so-called intensity of stress in the extreme fibre. It is 
difficult to state whether the ice failed by tension or com- 
pression. In some cases there was evidence of partial 
failure at least by internal shear. Some of these beams were 
placed so as to be loaded on their beds, so to speak, and some 
on edge, but on the whole there appeared to be little dif- 
ference in the results. Occasionally there appeared to be a 
tendency to fail in such manner as to exhibit the "bedding" 
planes. 

Table II 

BEAMS OF NATURAL ICE 
Load at Centre of Span 



Span. 


Number of Tests. 


Modulus of Rupture. Pounds per Sq. In. 




Max. 


Mean. 


Min. 


18 ins. 


9 


351 


247 


140 



BEAMS OF ARTIFICIAL ICE 



[8 ins. 



12 



138 



85 



There is the same inferiority of ultimate resistance of 
the artificial ice beams as in compression, but the artificial 
ice beams show a little less than half the modulus of rupture 
given by the natural ice beams. 



INDEX. 



Adhesion between bricks and stones 

and cement mortars, 373-375 
Adhesive shear or bond, 592-598, 633 
Alloys of copper, tin, aluminum, 

zinc in tension, 346-362 
Alloys of copper, tin, zinc in torsion, 

193-195, 546 

Alloys of copper, tin, zinc in beams, 

355, 561, 562 
Aluminum, 354, 355 
Aluminum alloys in bending, 560, 561 
Aluminum, alloys of, in tension, 

352-358 
Aluminum, alloys of, in torsion, 

193-196 
Aluminum-zinc beams, 354 
Angles, steel, as columns, 496-500 
Annealing of steel, 338, 339 



B 

Balanced economic steel reinforce- 
ment, 608-613, 617-618 

Batten plates, 508 

Beams of ice, 918 

Beams, solid, rectangular, and cir- 
cular, 554-562 

Bearing capacity of rivets, 441 

Bending and direct stress combined, 
254-267 

Bending and direct stress in eye-bars, 
255-267 



Bending and torsion combined, 246 
Bending moments and shears in 

general, 64 
Bending moments in concrete-steel 

beams, 614-616, 618-619 
Brass, 349-351 

Brick masonry beams, 584, 585 
Brick piers or columns, 413-417 
Bricks, adhesion between cement 

and, 373-375 
Bricks and brick piers in compression, 

409-419 
Bricks in s^'^earing, 550 
Bridge portal, stresses in, 789 
Briquette, Am. Soc. C. E. standard 

for cement tests, 372 
Bronzes and brass. Board of Water 

Supply, N. Y. City, 359, 360 
Building stones, 420 
Bulk modulus, 19 



Castings, steel, 322 
Cast-iron beams, 560 
Cast-iron columns, 520-527 
Cast-iron, elastic limit, 286 
Cast-iron, fatigue of, 295 
Cast-iron flanged beams, 662-664 
Cast-iron in compression, 388 
Cast-iron in shearing, 544 
Cast-iron in torsion, 192-193, 544 
Cast-iron, modulus of elasticity, 
286-290, 294, 389 

921 



922 



INDEX. 



Cast-iron, remelting and continued 

fusion of, 294 
Cast-iron, tensile resilience of, 286- 

294 
Cast-iron, tensile strain diagram, 285 
Cast-iron, ultimate tensile resist- 
ance, 292, 294 
Cement in compression, 395 
Cement in tension, 362-377 
Cement mortar in compression, 395 
Cement mortar in tension, 362-377 
Chemical elements in steel, 343 
Chrome vanadium steel, 329-331 
Cinder concrete in compression, 

405-407 
Cinder concrete in tension, 365 
Circular cylinders, torsion of, 884 
Clavarino's formula, 48, 913 
Coefficient of elasticity, see Mod- 
ulus of elasticity. 
Collapse of flues, 774-778 
Column design, 505-520 
Columns of cast-iron, 520-527 
Columns of concrete, 408 
Columns of timber, 528-529 
Columns, long, 169 
Columns, long, wrought iron and 

steel, 490-520 
Combined bending and compression, 

268 
Combined bending and direct stress, 

254 
Combined bending and torsion, 246 
Common theory of flexure, 49, 99 
Common theory of flexure for beam 

of two materials, 156 
Common theory of torsion, 182-196 
Composite material, elastic action 

of, 749 

Compression, 385 

Compressive resistance of cast-iron, 

388 
Compressive resistance of steel, 

389-391 



Compressive resistance of wrought 

iron, 387, 388 
Compressive stress, 4, 385 
Concrete columns, 408 
Concrete columns, reinforced, 641- 

655 
Concrete beams, 575-583 
Concrete in compression, 395-409 
Concrete-steel, adhesive shear, 592- 

598 
Concrete-steel beams, 600-640 
Concrete-steel beams, design of, 

629-640 
Concrete-steel, modulus of elasticity, 

633 
Concrete-steel members, 588-658 
Concrete-steel theory, by common 

theory of flexure, 591-620 
Connections, 435-473 
Connections, pin, 470-473 
Connections, riveted joints, 435- 

470 
Continuous beams in general, 118 
Copper, alloys of, in tension, 346- 

362 
Copper in compression, 396 
Copper in shearing and torsion, 546 
Copper in tension, 347-349 
Copper, tin, zinc beams, 561, 562 
Copper, tin, zinc, lead, and alloys 

in compression, 391-395 
Copper, under repeated stress, 361 
Core method for general flexure, 

735-739 
Core surface or section, 732 
Cover plates for plate girders, 

length of, 708-710 
Crank shaft stresses, 247-253 
Criterion for greatest moment, 84 
Curved beams, 712-719 
Cylinders, thick hollow, 203-223, 

847 
Cylinders, thin hollow, 197-201 
Cylinders, torsion of, 853-892 



INDEX. 



923 



D 

Deflection due to shearing, 125, 153 
Deflection in oblique flexure, 745-749 
Deflection in terms of greatest fibre 

stress, 124 
Deflection of beams, 121, 126-131 
Deflection of rolled-steel beams, 

'677, 678 
Design of columns, 505-520 
Design of concrete-steel beams, 

629-640 
Design of concrete-steel columns, 

653-655 
Diagonal riveted joints, 469 
Diameter of rivets, 445 
Distribution of shear in beams of 

various sections, 60, 62 
Distribution of stress in riveted 

joints, 437 
Division of loading between concrete 

and steel, 655 
Driving and drawing spikes, 781-786 
Ductility, 286 
Ductility of wrought iron, 302 

E 

Eccentric loading of any surface, 

725-735 
Effect of chemical elements on steel, 

343 
Effect of low temperatures on steel, 

333-335 

Effect of shop manipulation on steel, 

339 

Efficiency of riveted joint, 454-461 

Elastic limit, 5, 282 

Elastic limit of wrought iron, 298 

Elasticity, i, 4 

Elasticity, modulus of, 4, 281 

Ellipse of inertia, 478, 480 

Ellipse of strain, 43 

Ellipse of stress, 26, 33 

Ellipsoid of strain, 42 



Ellipsoid of stress, 36, 40 

ElHptical cylinder, torsion of, 186-188, 

541, 863 
End shear in bent beams, 68 
Equilibrium and motion, equations 

of internal, 820-846 
Euler's formula, 169 
Expansion and contraction (thermal) 

of mortar, concrete, and stone, 377 
Eye-bars of steel, 314-327 
Eye-bars subjected to bending and 

tension, 255, 258, 263 



Fatigue of metals, 795-806 

Flanged beams, 659-682 

Flanged beams with equal flanges, 

665-682 
Flanged beams with unequal flanges, 

661-665 
Flat plates, square, rectangular, cir- 
cular, elliptical, 765-774 
Flexure, common theory of, 49 
Flexure by oblique forces, 175 
Flexure, general treatment by core 

method, 735-739 
Flexure of beams, 1 21-168 
Flexure of beams of two materials, 

156 
Flexure of curved beams, 712-719 
Flexure of long columns, 169, 175 
Flexure, theory of, general formulae, 

897-912 
Flow of solids, 809-819 
Flues, collapse of, 774-778 
Formula (column) of C. Shaler 

Smith for timber columns, 53 1 , 532 
Formulae for long columns, 493-505 
Fracture of steel, 343 
Fracture of wrought iron, 302, 303 
Freezing cements and mortars, effect 

of, 375-377 
Friction of riveted joint, 465 



924 



INDEX. 



General formulae of theory of flexure, 

99, 897^12 
Girders, design of plate, 683-708 
Gordon's formula, 474, 481-490 
Granites in compression, 421-425 
Graphical determination of bending 

moments, 160 
Greatest intensity of shearing stress, 

29, 36, 163 
Greatest stresses in beams, 162 
Gun-bronze, 346, 349, 392 



H 



Hardening and tempering of steel, 

336, 337 
Helical spiral springs, 750-760 
High extreme fibre stress in short 

sohd beams, 556 
Hollow cylinders, thick, 203-223, 847 
Hollow cylinders, thin, 197 
Hollow spheres, thick, 224, 892 
Hollow spheres, thin, 201, 202 
Hooke's Law, 2, 3 
Hooks, stresses in and design of, 

719-725 
Hoop tension, 204, 206 

I 

Ice in compression and flexure, 

915-918 
Inclination of neutral surface of 

beam, 122, 123 
Influence of time on strains, 805 
Intensity of stress, 3 
Intermediate and end shear in bent 

beams, 68 



Jaws of columns, design of, 510, 511 
Joints, pin connections, 470-473 



Joints, riveted, 435, 470 
Joints, welded, 470 



Lateral strains, 9 

Lattice bars, 506-508 

Latticed columns, 506-516 

Launhardt's formula, 801, 802 

Law, Hooke's, 2, 3 

Least work, method ofj 788 

Length of cover plates for plate 

girders, 708-710 
Limestones in compression, 421-425 
Limit of elasticity, 5, 282 
Lag-screws, resistance to pulling out, 

784 
Long colums, 169, 175, 474-506 
Long column formula, 493-505 ^ 



M 



Magnesium, 354, 355 

Magnesium alloys, 355 

Manganese steel, 344 

Marbles in compression, 421-425 

Method of least work, 788 

Moduli of elasticity, relation be- 
tween, II 

Modulus of elasticity, 4, 281, 552 

Modulus of elasticity for tension and 
compression in terms of shearing 
elasticity, 19, 20 

Modulus of elasticity for torsion, 
186, 187, 540-542 

Modulus of elasticity of alloy beams, 
562 

Modulus of elasticity of aluminum- 
zinc beams, 354 

Modulus of elasticity of cast iron, 
286-290, 294, 389 

Modulus of elasticity of concrete, 399 

Modulus of elasticity of steel, 303- 
308, 390 



INDEX, 



925 



Modulus of elasticity of timber in 

tension, 380 
Modulus of elasticity of wrought 

iron, 297 
Modulus of rupture in bending solid 

rectangular and circular beams, 

554-562 
Moisture in timber, effect of, 426, 427 
Moment, greatest, produced by con- 
centrations, 83-86 
Moment in cantilever, 126, 128 
Moment of inertia, general treatment 

of, 475-480 
Moment of single load at centre of 

span, 95, 129 
Moment of uniform load, 80, 81, 96, 

129 
Moment produced by concentrated 

loads, 83 
Moment produced by two equal 

weights, 76 
Moments and shears in bent beams, 

64 
Moments in ordinary continuous 

beams, 131-142, 144-152 
Moments tabulated for plate girders, 

89, 90 
Mortise holes, shearing behind, 786 
Motion, equations of, 820-846 

N 
Natural building stones, 420-425, 549 
Neutral axis, 51, 52 
Neutral axis, position of, in reinforced 
concrete beams, 605-608, 610, 611 
Neutral curve for continuous beair.s, 

132-155 
Neutral curve for special cases, 

126-132 
Neutral surface, shearing in, 61, 63, 

163 
Nickel steel, 319, 325-328 
Notation concrete steel beams, 600- 

602 



Oblique or general flexure, 739-749 
Orthogonal stresses, 43 
Oscillations, torsional, 886 



Pendulum, torsion, 890 

Permanent set, 286 

Phoenix-column section, 488 

Phoenix columns, tests of, 490-496 

Phosphor-bronze, 361 

Phosphor-bronze wire, 361 

Pine, white, in compression, 432-434 

Pine, yellow, in compression, 427-434 

Pin connections, 470-473 

Pitch of rivets, 446-453 

Pitch of rivets in flanges of plate 
girder, 698-702, 710, 711 

Plane spiral springs, 761-765 

Planes of resistance in oblique 
flexure, 739-745 

Plate girder, design of, 683-708 

Plates, carrying capacity of, 766-774 

Points of contraflexure, 136, 138, 
148, 152 

Poisson's ratio, 10 

Portland cement and cement mortar 
in tension, 362-377 

Portland cement concrete in com- 
pression, 395-409 

Portland-cement concrete in tension, 

362-377 
Principal moments of inertia, 477-480 
Principal stresses, 23, 24, 26, 27, 40 
Punching, drilling, etc., of steel, 339 



Rail-steel, 323 

Reactions for bridge floor beams, 74 
Reactions under continuous beams, 
112, 114, 118 



926 



INDEX. 



Rectangular cylinders, torsion of, 
869-883 

Reduction of resistance between 
ultimate and breaking point, 285 

Reinforced concrete columns, 641-655 

Resilience, 231 

Resilience of cast-iron in tension, 290 

Resilience of flexure, 233 

Resilience of steel in tension, 311, 312 

Resilience of tension and compres- 
sion, 232 

Resilience of torsion, 240 

Resilience of wrought iron, 299, 300 

Resilience of shearing, 236 

Resilience, total, due to direct 
stresses and shearing, 239 

Resisting capacity of ice, 915-918 

Riveted joints, 435-470 

Riveted joints, butt-joints with 
double cover plates, for steel, 
436-452 

Riveted joints, distribution of stress 

in, 437 

Riveted joints for trusses, 468-470 

Riveted joints in angles, 469 

Riveted joints, lap-joints, and butt- 
joints with single butt-strap, for 
steel, 436-448 

Riveted joints, tests of full-sized, 
454-464 

Riveted steel in shearing, 443, 451, 
460 

Rivets, bearing capacity of, 441, 460 

Rivets, bending of, 440 

Rivets, diameter and pitch of, 445 

Rivets, shear of, 443, 460 

Rivets, steel, 324, 451, 460 

Rollers, resistance of, 778-781 



Sandstones in compression, 420-425 
Section modulus, 55 
Set, permanent, 286 



Shear, first derivative of moment, 65 
Shear, greatest caused by uniform 

load, 79 
Shearing, behind mortise holes, 786 
Shearing, greatest intensity of, 29, 

36, 163 
Shearing, modulus of elasticity, 5, 

186, 191 
Shearing stress in beams, 57, 165-167 
Shearing stress and strain, 13, 185, 

186, 540 
Shearing in neutral surface of timber 

beams, 57, 165-167, 571-574 
Shearing, ultimate resistance, 543-551 
Shears in bent beams, 64, 68 
Shears, single load located at centre 

of span, 95 
Shears, tabulated for plate girders, 

89, 90 
Shears, uniform load on span, 97 
Short blocks, 386 
"Short" test specimens, 309, 310 
Shrinkage stresses in thick hollow 

cylinders, 213 
Silica sand, Portland cement, and 

mortar in tension, 370, 371 
Spheres, thick hollow, 224-230, 892 
Spheres, thin hollow, 201-202 
Spikes, driving and drawing, 781-786 
Spiral springs, helical, 750-765 
Spiral springs, plane, 761-765 
Spruce columns, 429-434 
Spruce in compression, 439-443 
Steel, 303 

Steel, annealing, 338 
Steel castings, 322 
Steel, change of elastic properties 

under repeated stresses, 342 
Steel, effect of high and low tem- 
peratures, 333-335 
Steel, effect of punching, drilling, 

reaming, and shop processes, 339 
Steel, effects of chemical elements, 

343 



INDEX, 



927 



Steel, elastic limit, 310, 311, 330, 390 

Steel eye-bars, 314, 316 

Steel, fracture of, 343 

Steel, hardening and tempering, 336, 

337 
Steel, in compression, 389-391 
Steel, in shearing, 545 
Steel, in torsion, 190-192, 545 
Steel, modulus of elasticity, 303-308, 

390 
Steel, nickel, 325-328 
Steel rails, 323 
Steel reinforcement acquires stress, 

592-598 
Steel reinforcement, economic or 

balanced, 608-613, 617, 618 
Steel, resilience of, 311, 312 
Steel rivets, 324 

Steel, rolled flanged beams, 669-682 
Steel shapes and plates, 315, 317, 319 
Steel short solid beams, 558 
Steel, ultimate tensile resistance, 305, 

333 
Steel wire, 320, 321 
. Stone beams, 586, 587 
Stones, natural, in compression, 

420-425 
Stones, natural, in shearing, 549 
Straight-line formula for columns, 

494-504 
Strain, i, 2, 4 

Strains, influence of time on, 805 
Stress, I, 2, 3, 4 
Stress, intensity of, 3 
Stress parallel to one plane, 2 1 
Stress-strain curve, 6 
Stress-strain curves for cast iron, 288 
Stresses at any point in beam, 162 
Stresses, expressions for tangential 

and direct, 820-826 
Stresses of tension and compression, 

resolution of, 7, 8 
Structural steel, classes of, 303 
Suddenly applied loads, 242, 243 



Temperature, effect of high, 334, 335 

Temperature, effect of low, 333 

Tempering of steel, 336, 337 

Tensile stress, 281 

Terra cotta and columns, 415, 416, 
419 

Tests of riveted joints, 454-464 

Tests of steel angle and other col- 
umns, 496-503 

Tests of wrought-iron Phoenix col- 
umns, 490-496 

Theorem of three moments, 102, 109, 
III, 114 

Theory of flexure, general formulae, 
99, 897 

Thermal expansion and contraction 
of mortars, concrete, and stone, 

377-379 
Thick hollow cylinders, 203-223, 847 
Thick hollow spheres, 224, 892 
Thin hollow cylinders, 197 
Thin hollow spheres, 201, 202 
Timber beams, 563-575 
Timber columns, 528-539 
Timber in compression, 426-434 
Timber in shearing and torsion, 

547-548 
Timber in tension, 379-383 
Tin, 347, 349, 546 
Tin, alloys of, 346-356 
Tobin bronze, 351-357, 394 
Tobin bronze in compression, 394, 395 
Tobin's alloy, 346-357 
Torsion, 182, 196, 540, 853 
Torsion, combined with bending, 

246-253 
Torsion, general observations, 

186-188, 885 
Torsion, greatest shear in circular 

sections, 186-188, 541, 885 
Torsion, greatest shear in elliptical 

sections, 186-188, 541, 865 



928 



INDEX. 



Torsion, greatest shear in rectangular 
sections, 186-188, 541, 880-883 

Torsion, greatest shear in triangular 
sections, 868 

Torsion in equilibrium, 182-196, 540, 

853 
Torsion of circular sections, 182-196, 

541,884 
Torsion of elliptical sections, 186-188, 

541, 863 
Torsion of rectangular sections, 186- 

188, 541, 869 
Torsion of triangular sections, 866 
Torsion oscillations, 886 
Torsion pendulum, 890 
Torsion (twisting) moment in terms 

of H.P., 188, 189 
Tresca's experiments, flow of solids, 

810 
Tresca's hypotheses, flow of solids, 

8n 

U 

Ultimate resistance, 285, 543 

Ultimate resistance affected by high 
and low temperature, 333-335 

Ultimate resistance affected by re- 
peated stressing, 361, 795-806 

Ultimate resistance of cast-iron in 
tension, 292, 294 

Ultimate resistance of steel in ten- 
sion, 303-346 

Ultimate resistance of wrought iron, 
295-303, 387 

V 

Vanadium steel, 328-333 

W 

Web reinforcement in concrete-steel 

beams, 620-629 
Weight of concrete, 372 



Welded joints, 470 

Weyrauch's formula, 801, 803 

White-oak columns, 529 

White oak in compression, 429, 432, 

434 
White-pine columns, 528-539 
White pine in compression, 432-434 
Wire, steel, 320, 321 
Wohler's experiments, 796-799 
Wohler's law, 795-796 
Work expended in producing strains, 

231 
Working stresses in concrete steel 

beams, 629 
Working stresses in concrete steel 

columns, 650 
Wrought iron, 295-303,. 387 
Wrought-iron bars, diagram of 

strains, 298 
Wrought-iron beams, 680-682 
Wrought iron, ductility and resilience 

of, 297-300 
Wrought iron, fracture of, 302 
Wrought iron in compression, 387, 

388 
Wrought iron, in shearing, 543 
Wrought iron, in torsion, 192, 543 
Wrought iron, modulus of elasticity, 

296, 387 

Wrought-iron, short solid beams, 557 

Wrought iron, ultimate resistance, 

and elastic limit, and yield point, 

297, 388 



Yield-point, 7, 284 
Yield-point of wrought iron, 297 
Yellow-pine columns, 528-539 
Yellow pine in compression, 427-434 



Zinc, 346-362, 546 

Zinc, alloysof, 193-195, 346-362, 546 










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